Asymptotic Equivalence of Another Recurrence: So I have the following recurrence relation:  $$f(n) = f(n-1) + f(\lceil n/2\rceil)+ 1$$  I already know that:  If:  $$g(n) = g(n-1) + 1$$  $$g(n) = O(n)$$  If:  $$g(n) = g(\lceil n/2\rceil) + 1$$  $$g(n) = O(\log(n))$$  If:  $$g(n) = g(n-1) + g(\lceil n/2\rceil)$$  $$g(n) = O(nlog(n))$)  Thus can I definitely conclude that:  $$f(n) = O(n \log(n) * n * \log(n)) = O(n^2 \log(n)^2)$$
Strange Recurrence: What is it asymptotic to? So I have the following recurrence relation for the growth rate of an algorithm:  $T(n)$ = time taken by algorithm to solve problem of size n:  $$T(n) = T(n-1) + T(\lceil(n/2)\rceil)$$  Clearly this does not obey any polynomial law but now my question is what does it obey?  I merely want an asymptotic equivalence.  My guess is subexponential since this is faster than polynomial but clearly slower than any exponential form.  What though is the question....
How to Disable USB Auto-mount I use VirtualBox on my MBP and often need to mount my USB flash drive inside the guest.  The problem I'm running into is that the host OS (OS X Mavericks) immediately mounts the USB flash drive when I insert it.  When I try to unmount it (gracefully, through Finder), OS X complains that the device is in use.  Further research into lsof reveals that Spotlight and AVG are doing their thing with the flash drive.  Rather than fiddling with individual app settings, I'd like to just modify the host OS behavior so that the USB device doesn't auto-mount.  Any suggestions? Java logical operator short-circuiting Can I set background image and opacity in the same property? How to batch watermark JPG files? I have around 15,000 JPG files that need to be watermarked. Is there any way to process these images en masse and add a watermark?  The images may be different sizes, I want the watermark placed in the center of the images. A situation came up in my session where a fighter used his Shield Master Shove feature to shove a Giant Crocodile that was currently restraining another PC using it's bite+grapple ability.  I did a quick check at the time and did not find anything related to shove, although in the  condition description it states:     The condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when a creature is hurled away by the thunderwave spell.   I ruled at the time that a shove would count in a similar way to being "hurled away" however I wasn't sure and said I would find more concrete information for the future.  I have searched high and low for somebody else asking this specific question but have yet to find it. I saw some reference to a Jeremy Crawford tweet about involuntary movement and grappling however I am unable to find it.  If a third party "shoves" a creature that is grappling another, does it only shove the grappler away, or does it move both the grappler and grappled?  I drew a diagram:   (; this question relates to if a third party shoves the grappler instead, so it's not a duplicate.) When does Ubuntu 16.04 use /etc/apache2/ssl/apache.crt? I have some Ubuntu 16.04 server running and set up one vhost in apache2 which uses (or should use) ssl.  &lt;IfModule mod_ssl.c&gt;         &lt;VirtualHost *:443&gt;          ServerAdmin myemail@mailhoster.xx         ServerName my.domain.name.de         DocumentRoot /var/www/mysslsite          &lt;Directory /var/www/mysslsite&gt;                 Options FollowSymLinks MultiViews                 AllowOverride All                 Order allow,deny                 allow from all         &lt;/Directory&gt;          ErrorLog ${APACHE_LOG_DIR}/mysite.ssl.error.log         CustomLog ${APACHE_LOG_DIR}/mysite.ssl.access.log combined          &lt;/VirtualHost&gt; &lt;/IfModule&gt;   I did NOT enable the standard sites-available/default-ssl.conf.  I am able now to browse to  and it is asking in firefox to add some exception which is totally normal if i did not buy any certificate.  But I was wondering where it is set up that it should use the certificates in /etc/apache2/ssl/ directory. I can not find any config where it is told to use that. I all the time was thinking that it will use something from /etc/ssl folder.   Or did I maybe not activate SSL correctly? Showing $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$ Determining the characteristic of a field If $\mathbb{F}=\{ 0,1,x,y\}$ is a field with four elements, why is the characteristic 2?   In a field with four elements, the characteristic is 2 or 3(since it has to be a prime). Is there any way to determine precisely the characteristic of this field, and in general any finite field? How can I show a supremum of a set is also its limit point? It just seems like I can't wrap my head around this. For a set $A$ how can I show that the supremum of that set $\sup(A)$ is also a limit point of said set? Can I just set $$a=\sup(A)$$ $$b\le a \: \:  \forall \: b \in A$$ and then show that there's a point different from $a$ in any $\epsilon$-neighbourhood for any $\epsilon&gt;0$? Or should I use a completely different approach? how can i display the latest items from another site collection I have a web application, with three site collections . and i want inside my team site to be able to display the latest pages that  have been added to my enterprise wiki site collection. so what are the available approaches which i can follow?  Thanks in advance for any help.  Regards What is "umask" and how does it work? I believe that umask is something that controls , but do not fully understand it.  After running umask 0644 in a , I cannot read the files I create with the command-line text editor nano. I noticed that the permissions of that file are set to 0022 instead of the default 0755.  How does umask work? I thought I could just remove the each digit in the umask from 0777, 7 - 6 = 1 and 7 - 4 = 3, so I expect the permissions to be 0133, but apparently, this is not the case.   What is umask exactly? Explain it to me like I was a "Linux noob" How do I calculate with umask? What are use cases for umask? Is it possible to make my laptop's screen act as a second monitor for my desktop computer? How to change certain pages into landscape/portrait mode How can I change some pages into document into landscape mode, while leaving the others in portrait mode (or vice versa)? Order of nested random effects in lme4 I just fit a model in lme4, and I'm wondering what the heck I fit...  I have individuals id, and each is measured pass/fail on items that can be described using two factors, f1 and f2. My theory says that id and f2 can interact. So I want to compare these models:  # f2 doesn't matter m1 &lt;- lmer(pass ~ f1 + (1|id), family="binomial")  # f2 is a fixed effect, doesn't interact with individuals m2 &lt;- lmer(pass ~ f1 + f2 + (1|id), family="binomial")  # f2 is a nested random effect, interacting with individuals m3 &lt;- lmer(pass ~ f1 + (1|f2/id), family="binomial")   First, am I right that there's a random effect of each level of f2, and each individual's level of f2 is partially pooled, depending on the number of observations for that individual?  Second, does it make sense to fit f2 as a fixed effect with random individuals nested within each level of f2? How does one write that?  Third, what's this, with f2 and id reversed in the formula? I typed it by accident, but now I want to understand it.  m3weird &lt;- lmer(pass ~ f1 + (1|id/f2), family="binomial") How do question bookmarks work? Black nontransparent glass (Cycles) I am modeling plane and I don't want to model a cockpit so I want to use black windows which will reflect a lot of light but you can't see thought.   Currently I just use Glossy BSDF (GGX) with almost black color (0.01, 0.01, 0.01) and Rougness 0.015 but it don't look quite right. It lacks glass shine and looks too matt.  Edit:  Here is my simplified blend file:    Also here is what i want. Glass on this tie fighter is blackish and it looks like glass. In my blend file it looks more like plastic.    One more thing i noticed is when light reflect from glass whole glass section reflect same amount of light. It can be seen on tie fighter example above. What is PECS (Producer Extends Consumer Super)? How do I set an environment variable on the command line and have it appear in commands? Repeating "to" and "will" in enumerations of verbs Should I use the second "will" in constructions like this one: "it will definitely help you and will make the text more readable"  And should I write "to" before every infinitive in enumeration, or only before the first one, e.g. "it helps to develop and test" or "it helps to develop and to test"? Could this example be influenced by the fact that "help" can go with to-invinitive and bare infinitive, and with other verbs the rule would be different? How to move and overwrite subdirectories (and files) to parent directory?