I'm reading Kenneth Falconer's "Fractal Geometry" and it mentions on page 35 that box-counting dimension is not finitely stable. That is, in general it is not always the case that  $$ \dim_B (E \cup F) = \max (\dim_B E, \dim_B F)$$  But rather, this property is only true for the upper box-counting dimension (box-counting dimension defined by taking the upper limit).  I've been trying to think of a counter example to see why this is, but I can't think of one. Is there any simple example? Or does it require complicated constructions to show it's not true in general.
Here the lower box dimension and the upper box dimension are exactly what Falconer talks about in his book "Fractal Geometry". I already know that the upper box dimension is finitely stable since $N_\delta(A\cup B)\le N_\delta (A)+N_\delta(B)$, where $N_\delta(A)$ is defined as usual, and Falconer gives an example to show the box dimension may not be countably stable. (We know that, however, the Hausdorff dimension is countably stable).  But now I want to find the example to show the lower box dimension may not even be finitely stable.It seems that we can use von Koch curve to illustrate this, but what is the explicit explanation for this? Since the lower box dimension is monotonic, we need to find set A and B and show that   $\underline{\dim}_B(A\cup B)&gt;\max\{\underline{\dim}_B(A),\underline{\dim}_B(B)\}$.
I read on  that the netbook edition of Ubuntu was discontinued and that it is now integrated into the main distribution (as of 11.04).  I tried 11.04 on my new Samsung NC110-A06AU with 10.1" 1024x600 display. Most things worked out of the box, but some (like backlight power control) needed some  before they began to co-operate.  One thing that seems to be an ongoing problem is the resolution.  At 1024x600, most of the dialogue boxes run off the bottom of the screen, including the update manager. I have tried 11.10 beta 1 and it has the same problem.  Is there any way to fix this, or at least make it more usable? (I suspect not). Given the recent rise in popularity of netbooks, do you think Ubuntu will include a low-resolution-friendly version in the near future?  How well does KDE deal with these low resolutions? I noticed KDE Plasma has a netbook interface. Is it worth switching for that? Find the sum : $\frac{1}{\cos0^\circ\cos1^\circ}+\frac{1}{\cos1^\circ \cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+......+$ Find the sum of the following :   (i) $$\frac{1}{\cos0^\circ \cos1^\circ}+\frac{1}{\cos1^\circ\cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+......+\frac{1}{\cos88^\circ \cos89^\circ}$$  I tried :  $$\frac{1}{\cos1^\circ}\left[\frac{\cos(1^\circ-0^\circ)}{\cos0^\circ\cos1^\circ} + \frac{\cos(2^\circ-1^\circ)}{\cos1^\circ\cos2^\circ}+...\right]$$  = $$\frac{1}{\cos1^\circ}\left[\frac{\cos1^\circ\cos0^\circ}{\cos0^\circ\cos1^\circ} - \frac{\sin1^\circ \sin0^\circ}{\sin0^\circ\cos1^\circ} + \frac{\cos2^\circ \cos1^\circ}{\cos1^\circ\cos2^\circ} -\frac{\sin2^\circ \sin1^\circ}{\cos1^\circ\cos2^\circ}...\right]$$  For this, as well, I am not getting any pattern to solve further. Please suggest, thanks. How to replace all occurrences of a string in JavaScript? How can I list references from BibTeX file in-line with commentary? I would like to take the full citation of a reference in my BibTeX file and write commentary in-line after it.  A \fullcitation{id} command would be perfect.  I would like to use it as:  \fullcitation{id1}   This paper talks about A with applications to B.  \fullcitation{id2}   This paper considers how C affects A.  Whenever the \fullcitation{id} command appears, I want it to be replaced with the full reference, including authors, title, journal, and date.  Is this possible? I need to factory reset my Macbook but cannot recall the password. So I used Cmd+R during startup and go to terminal and typed 'resetpassword' according to this advice    Now 'Reset Password' application appears but there is nothing where I have to select the volume containing the user account. Why does this happen? Showing that $G$ is a group under an alternative operation. Anime with a boy hiring a creature from a stone, meets a man named Dante and starts a journey to collect crystals What causes the portraits of Wizards to behave interactively? I recently wiped my computer to have 14.10 as my only OS. It has 160 GB of storage. That was a month ago. Now, I keep getting prompts that the storage is almost full. And when I look it says I only have 543 MB free. How is this possible? I haven't torrented any large files. I do have Variety to autodownload wallpapers and not erase though.  Filesystem Size Used Avail Use% Mounted on  /dev/sda1  146G  138G  554M  100%  /  none 4.0K 0 4.0K 0% /sys/fs/cgroup  udev 742M 4.0K 742M 1% /dev  tmpfs 151M 1.1M 150M 1% /run  none 5.0M 4.0K 5.0M 1% /run/lock  none 752M 540K 751M 1% /run/shm  none 100M 80K 100M 1% /run/user –  er – This might seem like a very simple question, but what is the easiest way to turn a normal vertical list into a horizontal one? Why this polynomial is irreducible? Let $K=\mathbb{Z}_p(t)$. How to prove $f(x)=x^p-t$ is irreducible in $K[x]$? Can't see a glass object through another glass object in Eevee I was just testing Eevee and found that you cannot view a glass object through another glass. Is there any workaround? It works fine only If I rotate the glass window or put my camera to the back side of the scene Take for example  both at SO and SU. Now that there's , (new) users asking a LaTeX question should at least be informed that there is a new Stack Exchange site dedicated to TeX and thus increases their chance of an answer. This should happen before the question is actually posted and ideally offer the option to post the question directly there instead (without the need to copy-paste).  LaTeX is just an example, there's also , ,  etc., and it is obvious that this can only be decided on a case-by-case basis which is why I certainly do not suggest migrating questions automatically when these tags are used. But informing users of that possibility would probably help the SE network in general. How can a "proper" function have a vertical slope? Plotting the function $f(x)=x^{1/3}$ defined for any real number $x$ gives us:   Since $f$ is a function, for any given $x$ value it maps to a single y value (and not more than one $y$ value, because that would mean it's not a function as it fails the vertical line test). This function also has a vertical tangent at $x=0$.   My question is: how can we have a function that also has a vertical tangent? To get a vertical tangent we need 2 vertical points, which means that we are not working with a "proper" function as it has multiple y values mapping to a single $x$. How is it possible for a "proper" function to have a vertical tangent?  As I understand, in the graph I pasted we cannot take the derivative of x=0 because the slope is vertical, hence we cannot see the instantaneous rate of change of x to y as the y value is not a value (or many values, which ever way you want to look at it). How is it possible to have a perfectly vertical slope on a function? In this case I can imagine a very steep curve at 0.... but vertical?!? I can't wrap my mind around it. How can we get a vertical slope on a non vertical function? Wait until all jQuery Ajax requests are done? View referrals from one part of my site to another in Google Analytics I have an online shop with a blog structured this way:   Shop: example.com Blog: example.com/blog   Both index have the same Google Analytic Code (UA). In the blog I write articles about some products with a link to the product page.  Now the problem is, I want to see stat on visits whose referral links have this pattern example.com/blog/*. Having changed the scene units and unit scale, how can I change the default size of new objects? I am using blender most of the time to model things for my 3d printer. So a default unit in "meters" is not really helpful!  Within the scene settings it is simply possible to set the units to millimeter and the unit scael to 0.001 so everything seems to work as I want it to work...   But if I add a new primitive it uses its old default sizes in meters... so a new cube get's 2000 x 2000 millimeters instead of 2x2 or whatever will be useful..  Same for all other new objects...  How can I change the default sizes of new objects permanently?   And how can I change the settings in the application templates? Right now using ctrl+u changes the "general" template - perhaps because I loaded it before...   Thanks a lot and best regards Stefan I have this function:  $$ f(x,y) = \frac {xy}{|x|+|y|} $$  And I want to evaluate it's limit when $$ (x,y) \to (0,0)$$ My guess is that it tends to zero. So, by definition, if:  $$ \forall \varepsilon \gt 0, \exists \delta \gt 0 \diagup \\ 0\lt||(x,y)||\lt \delta , \left|\frac{xy}{|x|+|y|}\right| \lt \varepsilon $$ Then $$ \lim_{(x,y)\to(0,0)}\frac {xy}{|x|+|y|} = 0 $$ So:  $$ \left|\frac{xy}{|x|+|y|}\right| = \frac{|xy|}{|x|+|y|} = \frac{|x||y|}{|x|+|y|} \le 1 |y| \lt \delta $$  So for any $$\delta \lt \varepsilon$$ the inequality is true. Hence, the limit exists and is equal to zero.  . Am I wrong or is Wolfram|Alpha wrong? Cleaning Contaminated Disc Rotors I've had an eternal struggle with my current bike over brake contamination. The levers and calipers are shimano ultegra r8000, so the latest and (almost) greatest.  I first noticed it when I took it to a repair shop (i no longer use) for a bleed, it came back and the front brake would squeal like crazy and at low speeds had little to no braking force. You can stop from speed but then struggle to come to a halt without using your foot.  At around 2000 miles I tried to replace the brake pads, when I took them out they were black with grime on both sets (front &amp; back), not to mention worn down to the metal. I replaced them with the L04C sintered pads, gave the callipers a good cleaning with a towel (no chemicals) and fitted the new set.  Annoyingly all of my braking issues are back, now I probably should have cleaned the rotors, I simply used some muc off disc brake cleaner on them. I would like to know the best way to clean the rotors, I've inspected the pads and they are looking fairly black now, with plenty of tread left. I'm going to abandon these pads and fit a new set. For the rotors I plan to completely remove them from the wheels (they are center lock fitted) and try and treat them off the bike, what is the best way to fully clean them? They have plenty of wear left in them but you can see these black grooves in them, and plenty of muck on the back that I struggle to get off.  Below are images of the rotors after soaking them in disc brake cleaners, waiting and then wiping them down with a clean, microfiber towel.   Front Wheel Outside Rotor   Front Wheel Inside Rotor - note the increased dirt on this side  Looking at ways people clean their disc rotors, I've seen plenty of suggestions that I would need to burn the contaminants off the discs. Would the best approach be to remove them and use a blowtorch, should I soak them in some nasty chemicals overnight before I try to heat them? Do I try baking them in the oven?  And finally, what is the best way to prevent the brakes from becoming contaminated in the future? it's really off putting when they squeal and don't perform how I'd expect because it means I'm more reserved whilst out riding on the bike, leading to reduced time riding around outside. I commute on this bike (50 miles a week) and then ride about 50 on the weekends also, so it see's it's fair share of road use. I tend to wash it once after my big ride on saturdays and will rinse it off if it's been raining, any day that occurs (since I live in north east england, the acid rain..). I've owned disc brake bikes before and never had half the issues I've had with this, I had a cheap trek with tektro brakes and the brakes were sharper and never had any issues with contamination whatsoever (i did around 500 miles on that bike before a motorist happened).  Thanks for any and all advice :) Why is the sun brighter in Australia compared to parts of Asia?