Problem 1.58 in  ask us to find the dimension of the Lie algebra generated by $x$ and $y$ with defining relations $[x,[x,y]]=0$ and $[y,[y,[y,[y,x]]]]=0$.  I've managed to show that the algebra is spanned by the six elements $x$, $y$, $z$, $w$, $v$, $u$ with $[y,x]=z$, $[y,z]=w$, $[y,w]=v$, $[z,w]=[v,x]=u$, and all other brackets equal to $0$. But to show that the algebra has dimension 6, and not less than 6, I need to show that these elements are linearly independent. My strategy for that is to find six linearly independent matrices that satisfy those commutation relations, but I can't for the life of me figure out how.  Is there any good way to find matrices that satisfy those relations, or is there a better strategy to tackle this problem?   addresses part (b) of the same problem, but unfortunately it doesn't help here - when we change $ad(y)^4$ to $ad(y)^5$ the structure changes significantly (in particular, it becomes infinite dimensional), and that question doesn't cover any general methods.
This is a problem from Introduction to Representation Theory by Etingof et al. I have no idea how to approach it. My friend wrote a program that uses Gaussian elimination to find a basis, which starts with x, y, [x, y], [y, [x, y]], ... but freezes once it gets to elements with 9 or more occurrences of x, y.  I expect that there is some way to characterize the nonzero elements of this Lie algebra (i.e., elements that are "immune" to being shown to be zero through repeated application of the Jacobi identity) which could be turned into a way to enumerate them. I don't see what it is, though.  It's also totally unclear to me how to show that some set we know spans the Lie algebra (proven using some criterion as described in the previous paragraph) is actually linearly independent, unless we find a faithful representation... but I have no idea how to find a faithful representation for an infinite-dimensional Lie algebra.  Hints would be greatly appreciated.  Edit: After looking at the partial output, we can form the "straightforward" conjecture that:   If $n \geq 4$ is even, then there is one basis element of grade $n$, namely $\mathrm{ad}([x, y])^{(n-4)/2}[y, [y, [y, x]]]$. If $n \geq 5$ is odd, then there are two basis elements of grade $n$, namely $[x, z]$ and $[y, z]$ where $z$ is the single basis element of grade $n-1$.   I also have a different conjecture, which is mutually exclusive with the above: that if we flatten out all nonzero elements into the form ${\rm ad}(e_1) {\rm ad}(e_2) \ldots {\rm ad}(e_{n-1}) e_n$ using the Jacobi identity, then in the sequences $e_1, \ldots, e_{n-1}, e_n$ and $e_1, \ldots, e_n, e_{n-1}$, any span containing two x's must contain at least two y's and any span containing five y's must contain at least two x's, otherwise we can use the Jacobi identity to rearrange the element somehow to show that it vanishes. If this is true, then the long-range behaviour must be:   For $n \geq 8$ there is exactly one basis element of grade $n$, namely $[... [y, [x, [y, [y, [x, y]]]]] ...]$ (the pattern $y x y$ repeats).   I still don't see an approach to proving that none of the elements in this sequence actually vanishes, though.  Edit 2: Here is the hint given by Etingof in an email:     No, there are no published solutions, since these problems are used as   a homework for classes. This is indeed a nontrivial problem.      This is the positive part of the affine Kac-Moody algebra of type   A_2^2, which is the twisted loop algebra for sl(2). So what you need   to do is to relate this Lie algebra to loops into sl(2). I see from   your post that you are on the right track.
How to enable top bar background opacity? There is a way to change top bar background opacity in ubuntu 20.04 lts.  I installed the hide top bar extension but is not a option to change it. In Bishop's book "Pattern Classification and Machine Learning", it describes a technique for regularization in the context of neural networks. However, I don't understand a paragraph describing that during the training process, the number of degrees of freedom increases along with the model complexity. The relevant quote is the following:     An alternative to regularization as a way of controlling the effective   complexity of a network is the procedure of early stopping. The   training of nonlinear network models corresponds to an iterative   reduction of the error function defined with respect to a set of   training data. For many of the optimization algorithms used for   network training, such as conjugate gradients, the error is a   nonincreasing function of the iteration index. However, the error   measured with respect to independent data, generally called a   validation set, often shows a decrease at first, followed by an increase as the network starts to over-fit. Training can therefore be   stopped at the point of smallest error with respect to the validation   data set, as indicated in Figure 5.12, in order to obtain a network   having good generalization performance. The behaviour of the network   in this case is sometimes explained qualitatively in terms of the   effective number of degrees of freedom in the network, in which this   number starts out small and then to grows during the training process,   corresponding to a steady increase in the effective complexity of the   model.   It also says that the number of parameters grows during the course of training. I was assuming that by "parameters", it refers to the number of weights controlled by the network's hidden units. Maybe I'm wrong because the weights are prevented to increase in magnitude by the regularization process but they don't change in number. Could it be referring to the process of finding a good number of hidden units?  What's a degree of freedom in a neural network? What parameters increase during  training? Positive operator has a positive spectrum? Let $T : \operatorname{dom}(T) \rightarrow H $ be a positive self-adjoint operator, is it then true that $\sigma(T) \subset [0,\infty)$? This is something that sounds natural and I guess that it is true. For bounded operator, this can be easily shown, but the only proof I know cannot be adapted to unbounded operator, so I was wondering if anybody here knows how to show this? I have a script which calls two commands:  long_running_command | print_progress   The long_running_command prints progress but I'm unhappy with it. I'm using print_progress to make it nicer (namely, I print the progress in a single line).  The problem: Connection a pipe to stdout also activates a 4K buffer, so the nice print program gets nothing ... nothing ... nothing ... a whole lot ... :)  How can I disable the 4K buffer for the long_running_command (no, I do not have the source)? Wrong favicon displayed on Google's mobile SERP For some time now, Google has been displaying a website's favicon next to its title on the mobile SERP according to    But, for one website that I am managing, it is showing a completely different favicon. You can see it if you open the  on a mobile device, I am talking about the orange one:  I am referring to the top result.  As you can see on the , the favicon has been added correctly using the proper meta tag:   &lt;link  rel="icon" type="image/x-icon" href="https://www.hoofgroove.ch/pub/media/favicon/default/favicon.ico" /&gt;   This is according to .  Also, all the browsers are displaying the correct favicon, which is not the orange one but the black&amp;red one.  This meta tag has been used for weeks and also the  has always been the correct one as far as I can tell.  But for some reason, Google is showing the wrong favicon on its mobile SERP.  Any ideas why? In the construction of types of numbers, we have the following sequence:  $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$  or:  $$2^0 \mathrm{-ions} \subset 2^1 \mathrm{-ions} \subset 2^2 \mathrm{-ions} \subset 2^3 \mathrm{-ions} \subset 2^4 \mathrm{-ions} $$  or:  "Reals" $\subset$ "Complex" $\subset$ "Quaternions" $\subset$ "Octonions" $\subset$ "Sedenions"  With the following "properties":   From $\mathbb{R}$ to $\mathbb{C}$ you gain "algebraic-closure"-ness (but you throw away ordering). From $\mathbb{C}$ to $\mathbb{H}$ we throw away commutativity. From $\mathbb{H}$ to $\mathbb{O}$ we throw away associativity. From $\mathbb{O}$ to $\mathbb{S}$ we throw away multiplicative normedness.   The question is, what lies on the right side of $\mathbb{S}$, and what do you lose when you go from $\mathbb{S}$ to one of these objects ? Google Docs thumbnails do not update, can I force an update? How do I find the package that provides a file? Simple enough question: is there some shell command (or GUI method) I can use that, given the path to a file on my system, tells me what package put it there? Assuming the file did in fact come from a package, that is.  Bonus question: what if it's a file that isn't installed on my system? Is there, say, a website that will let me look up a file and see what packages, if any, provide it? Some GUI apps use OS X location services, but I want to retrieve a Mac's physical location from the . This could be useful for running scripts, switching settings etc. The mechanism should be , not a Geo IP service (data too low-res, needs online connection).  Minimum requirement is output of longitude and latitude. Nice would be a number indicating the precision. Super nice: a geocoder that provides location names. Deployed web parts don't show up in Web Part Gallery Does getting disconnected from an SSH session kill your programs? Is diamagnetism a static or dynamic effect? When we put a diamagnetic material in the presence of an external magnetic field $\vec B_0$, the magnetic field inside the material decreases to $$\vec B=(1+\chi_m)\vec B_0,$$ where the magnetic susceptibility $\chi_m$ is a small negative number. I am assuming the material is linear and isotropic.  On the other hand the diamagnetism is explained in terms of Lenz law. When we change the magnetic flux of the external field over the material, atomic currents generate an induced magnetic field trying to restore the flux. But then the induced field could have any sign (in the appropriated direction), depending if we are increasing or decreasing the flux of the external field. It seems diamagnetism is a dynamic effect. How come the sign of $\chi_m$ is always negative? Moreover, the magnitude of $\chi_m$ should depend on how large is the variation of the flux but I do not see any suggestion of this when looking at tables of $\chi_m$. Using Moq to determine if a method is called It is my understanding that I can test that a method call will occur if I call a higher level method, i.e.:  public abstract class SomeClass() {         public void SomeMehod()     {         SomeOtherMethod();     }      internal abstract void SomeOtherMethod(); }   I want to test that if I call SomeMethod() then I expect that SomeOtherMethod() will be called.   Am I right in thinking this sort of test is available in a mocking framework? What is the exact $SU(2)$ representation to which these Hermitian generators belong? \begin{equation} t_a=\{t_1,t_2,t_3\}=\left\{\frac{1}{\sqrt{2}}\begin{pmatrix}  0 &amp; 1 &amp; 0 \\ 1 &amp; 0 &amp; 1 \\ 0 &amp; 1 &amp; 0 \end{pmatrix},\,\frac{1}{\sqrt{2}}\begin{pmatrix}  0 &amp; -i &amp; 0 \\ i &amp; 0  &amp; -i \\ 0 &amp; i &amp; 0 \end{pmatrix},\,\begin{pmatrix}  1 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; -1  \end{pmatrix} \right\} \end{equation}  I'm a bit puzzled with this, these are not the generators of the triplet representation of $SU(2)$ (the triplet in $SU(2)$ is the adjoint rep which is real and constructed with the structure constants), nevertheless they are used as if they were in much literature. What are really these generators? They seem like the extension to 3 dimensions of the Pauli matrices. The 3 dimensional fundamental rep (this does not make sense to me)? some kind of non-irreducible representation? What are good basic statistics to use for ordinal data? I have some  gained from survey questions.  In my case they are  responses (Strongly Disagree-Disagree-Neutral-Agree-Strongly Agree).  In my data they are coded as 1-5.  I  don't think means would mean much here, so what basic summary statistics are considered usefull? I have been trying to create a function that does basically the same thing that the QGIS "dissolve" function. I thought it would be super easy but well apparently not. So from what I have gathered around, the use of fiona with shapely should be the best option here. I just began to mess around with vector files so this world is pretty new to me and python as well.  For these example, I am working with a county shapefile founded here  So here are some piece of code I gathered but can't find a way to make them work together  For now my best method is the following based on : . It works fine and I get a list of the 52 states as Shapely geometry. Please feel free to comment if there is a more straight forward way to do this part.  from osgeo import ogr from shapely.wkb import loads from numpy import asarray from shapely.ops import cascaded_union  ds = ogr.Open('counties.shp') layer = ds.GetLayer(0)  #create a list of unique states identifier to be able #to loop through them later STATEFP_list = [] for i in range(0 , layer.GetFeatureCount()) :     feature = layer.GetFeature(i)     statefp = feature.GetField('STATEFP')     STATEFP_list.append(statefp)  STATEFP_list = set(STATEFP_list)  #Create a list of merged polygons = states  #to be written to file polygons = []  #do the actual dissolving based on STATEFP #and append polygons for i in STATEFP_list :      county_to_merge = []     layer.SetAttributeFilter("STATEFP = '%s'" %i )      #I am not too sure why "while 1" but it works      while 1:         f = layer.GetNextFeature()         if f is None: break         g = f.geometry()         county_to_merge.append(loads(g.ExportToWkb()))      u = cascaded_union(county_to_merge)     polygons.append(u)  #And now I am totally stuck, I have no idea how to write  #this list of shapely geometry into a shapefile using the #same properties that my source.   So the writing is really not straight forward from what I have seen, I really just want the same shapefile only with the country dissolve into states, I don't even need much of the attribute table but I am curious to see how you can pass it on from the source to the new created shapefile.  I found many pieces of code for writing with fiona but I am never able to make it work with my data. Example from  :  from shapely.geometry import mapping, Polygon import fiona  # Here's an example Shapely geometry poly = Polygon([(0, 0), (0, 1), (1, 1), (0, 0)])  # Define a polygon feature geometry with one attribute schema = {     'geometry': 'Polygon',     'properties': {'id': 'int'}, }  # Write a new Shapefile with fiona.open('my_shp2.shp', 'w', 'ESRI Shapefile', schema) as c:     ## If there are multiple geometries, put the "for" loop here     c.write({         'geometry': mapping(poly),         'properties': {'id': 123},     })   The problem here is how to do the same with a list of geometry and how to recreate the same properties than the source. Are there some good overviews of basic facts about images and inverse images of sets under functions? Can we re-request features that were declined a sufficient while ago? How to understand the LaTeX warning “there were undefined references”? How to Achieve this Requirement ???Can i use Jave Script , CSS and Visual force page?