Is $L^p$ space separable?
Space $\mathcal{L}^p(X, \Sigma, \mu)$ is separable iff $(\Sigma, \rho_\Delta)$ is separable
(used to show two posts from MSE)   (there used to be a list of the accounts)   (it was there) seems to be fixed A word used to describe someone who loves to be photographed What is a referee in the manuscript submission process? Please submit, with the manuscript, the names, addresses (including   countries), and e-mail addresses of five potential referees who are   outside the authors' institutions and must not have conflict of   interest with the authors.   Do you think I should name 5 researchers from a similar area that are not affiliated with my university and co-authors? Is this what referees are? Why didn't Escobar's hippos introduced in a single event die out due to inbreeding Today I read a  about how Pablo Escobar had once imported 4 hippos (1 male, 3 female) into his estate in Colombia for his private zoo. After his downfall, while other species were shipped out, hippos were considered too big to move and expected to not survive. However, to the surprise of all the hippos are thriving and are so numerous that there have been calls to cull them. From the report:  Numbers are projected to only get bigger. [Colombian biologist Nataly] Castelblanco and her peers say the population will reach over 1,400 specimens as early as 2034 without a cull - all of them descended from the original group of a male and three females. In the study, they envisaged an ideal scenario in which 30 animals need to be culled or castrated every year to stop that happening.  My understanding is that since there was only 1 male, the gene pool would be limited and lead to lot of inbreeding in the descendants. This would cause population to not explode because some individuals would be unfit to survive. Why has this not happened in case of hippos? Is it because there are 3 females (probably unrelated to each other) which keeps gene pool large enough? Or can mutations explain this phenomenon? Would results have been different if originally 2 females had been moved and only 1 retained? EDIT - I have just found that  is maybe less than 100 individuals which though big is not massive. EDIT2 - I have edited the question title to keep focus on the hippos although a general answer would be welcome. Posting images in chat using drag and drop I think it will be a good idea if there is a drag and drop image posting in chat. User needs to follow many steps to upload an image using upload button. The square root of a prime number is irrational What is the easiest way to swap occurrences of two strings in Vim? What's the easiest way to install a missing Perl module? I noticed that after joining a table to a shapefile the headers of each fields in the table used for the join are changed by QGIS and they are attributed to a different name.  For instance, I had a shapefile with vegetation_codes and a table with vegetation_codes and veg_definition (these were the headers). After joining the table to the shapefile and also saving the shapefile and creating a new one, the vegetation_classes header was changed into vegetati_1.  Is there a way to keep the original headers?  I am using QGIS Dufour on Windows 7 and the table that I join to the shapefile is CSV (tried with dbf as well and the problem is the same). Proving this relation is transitive Let $r$ be a relation on $A \times A$ such that $(a,b) r (c,d) \iff ad = bc.$ How can I show that this relation is transitive, ie. $(a, b)r(c,d)$ and $(c,d)r(e, f) \implies (a,b)r(e,f)$?  I tried to say that $(a,b) r (c,d)$ means that $c=ka$ and $d=kb,$ $k$ a coefficient, such that $ad = bc \iff akb = bka,$ and going on from there, but I'm not sure this is valid for all values of $a, b, c,$ and $d$. How can I show the transitivity of the relation? Storage disk format to use to be utilised by all OS In PCA, when the number of dimensions $d$ is greater than (or even equal to) the number of samples $N$, why is it that you will have at most $N-1$ non-zero eigenvectors? In other words, the rank of the covariance matrix amongst the $d\ge N$ dimensions is $N-1$.  Example: Your samples are vectorized images, which are of dimension $d = 640\times480 = 307\,200$, but you only have $N=10$ images. How can I store and retrieve a custom list in a blend file? I am trying to create a script that will allow me to generate a list and store it in the blend file. The list starts out blank and then strings get added to the list, resulting in: [&quot;test1&quot;,&quot;test2&quot;,&quot;test3&quot;] I create a PropertyGroup Class for the enum, define a new scene property as a CollectionProperty type with access to that enum Class, before appending a new item into the enum: import bpy  class item_set(bpy.types.PropertyGroup):     myItems = []     enums = bpy.props.EnumProperty(name=&quot;My Items&quot;, items=myItems) bpy.utils.register_class(item_set)  bpy.types.Scene.myItemSet = bpy.props.CollectionProperty(type=item_set)      newItem = bpy.context.scene.myItemSet.add() newItem.myItems.append(&quot;test&quot;)    I need to be able to add items to the list, save the blend file, re-open the file, run the script again and retrieve the list from the scene. However, the  states:  One thing to keep in mind is that after saving and reloading your blend-file, your new property will have disappeared...    ...To make the values you assigned in a previous session available again, you need to redefine the property. So after reopening the blend-file you need to rerun: bpy.types.Object.myProperty = bpy.props.StringProperty()   I can do the appending into and retrieval of the property values absolutely fine upon an initial run, but as the documentation states, the property is unavailable when trying to view it's contents on a reload of the file (after having run the script, saved and re-opened the file), but I am unsure how to redefine it without clearing all it's values. If I redefined the scene property, it would presumably also need the enum Class to be available, which would currently set myItems back to [] (empty). Do I need to change the way I am initially setting the items in the class or am I on entirely the wrong track with all of this? All I need to do is store a list in the blend file, so if there is a better way please let me know. Thanks. (I would hope, eventually, to be able to store sub lists [[&quot;a&quot;,&quot;b&quot;],[&quot;c&quot;,&quot;d&quot;],[&quot;e&quot;,&quot;f&quot;]], I'm just trying to get normal lists to work first.) How to SEO a Single-Page website I am creating a new website for a client, and I think a single page website / layout will be pretty neat. There is not that much detailed content, it is a company that imports a lot of products from different companies. So it will be a bit of a showcase of the different types of products / equipment you can order. Of the whole catalog only maybe 10 products are actually shown.  Now I have the basic structure with sections and links like:      #home / #products / #equipment / #services / #contact.    Now from what I understand is that Google does not index links containing #, so there will be no problem of duplicate content, everything will be indexed as the frontpage.  Now what I want to optimize is searching. This company is agent of some product in the country, meaning only they have the permission to import and sell that product. This means  that when somebody searches for product x, this website pops up as the first hit.  Now can I make it that they will be automatically brought to the #products part of the website directly from google.  I was thinking of something like having google index the different sections as different pages. Is something like that possible? What is the gradient of $f(x, y, z) = \sqrt{x^2+y^2+z^2}$?  I know that $\nabla f = \left\langle\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right\rangle$ or equivalently $\nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}$.  Eventually, I am going to evaluate this gradient at a point and then dot it with some given unit vector to find a directional derivative. I think maybe what is troubling me is the radical in $f$. Will I have to use the chain rule in order to find the three partial derivatives?  I tried this:  $$\frac{\partial f}{\partial x} = \frac{1}{2}(x^2 + y^2 + z^2)^{-\frac{1}{2}}2x$$ $$= \frac{x}{\sqrt{x^2 + y^2 + z^2}}$$  That result looks ugly to me. However, if I did that partial derivative correctly, then I think I know what I am doing. How to boot WUBI installed Ubuntu within Windows 8? How does a cell phone charger work? I have studied that there is a transformer which steps down the ac voltage after then bridge rectifier to convert into ac to dc and then given to filter, this is what I have learnt in books, but when I open a mobile charger I couldn't figure it out how it actually works? Can I turn off macOS Catalina's internal display without close the lid while using an external monitor? I  and am I sometimes find myself writing something like this:     XXX is a project I admire and am very interested in.   The "I &#60;verb> and am &#60;something>" feels strange here. It somehow sounds more natural in the third person: "He admires and is very interested in...."  Am I just imagining things – is it OK to use this construction, or should I use something completely different? Space between 2 paths in Illustrator How can I align 2 paths perfectly? No matter what I do I get a little space (gap between the 2 paths I create, and they are straight lines not as I saw in other questions with pixel problems)  An example:    Between every triangle is a little space, like in the example. It isn't obvious and it doesn't bother, but if I use the image (as an SVG or something) it is really obvious.