For every $n$ there exists $m$ such that $m/n$ is an upper bound but $(m-1)/n$ is not
Terence Tao, Analysis I, Ex. 5.5.2: Entry point needed
Which registry keys are impacted with Xerox Printer driver installation? I am implementing a driver deployment with Samba.  Every possible model of printer found an easy solution, except for Xerox Workcenter.    After browsing the Internet, I found that Xerox printer driver has to change some registry keys in order to work, keys which are changed by the executable downloadable on the manufacturer website.  Do you know which key are impacted by the Xerox Workcenter GPD driver?  This corresponds to the error message I get while uploading to Samba, which asks the position of a file named as a registry key.  I browse Samba bug tracker and they did not specify which registry key are impacted.  I contacted Xerox itself, but despite that fact that we have a support contract and really a huge number of workstations, it seems that Xerox is not willing to help us. Computing $\lim_{(x,y)\to (0,0)}\frac{x+y}{\sqrt{x^2+y^2}}$ What is the result of  $\lim_{(x,y)\to (0,0)}\frac{x+y}{\sqrt{x^2+y^2}}$ . I tried to do couple of algebraic manipulations, but I didn't reach to any conclusion.  Thanks a lot. Understanding MCMC and the Metropolis-Hastings algorithm Over the past few days I have been trying to understand how Markov Chain Monte Carlo (MCMC) works. In particular I have been trying to understand and implement the Metropolis-Hastings algorithm. So far I think I have an overall understanding of the algorithm but there are a couple of things that are not clear to me yet. I want to use MCMC to fit some models to data. Because of this I will describe my understanding of Metropolis-Hastings algorithm for fitting a straight line $f(x)=ax$ to some observed data $D$:  1) Make an initial guess for $a$. Set this $a$ as our current $a$ ($a_0$). Also add $a$ at the end of the Markov Chain ($C$).  2) Repeat the steps bellow a number of times.  3) Evaluate current likelihood (${\cal L_0}$) given $a_0$ and $D$.  4) Propose a new $a$ ($a_1$) by sampling from a normal distribution with $\mu=a_0$ and $\sigma=stepsize$. For now, $stepsize$ is constant.  5) Evaluate new likelihood (${\cal L_1}$) given $a_1$ and $D$.  6) If ${\cal L_1}$ is bigger than ${\cal L_0}$, accept $a_1$ as the new $a_0$, append it at the end of $C$ and go to step 2.  7) If ${\cal L_1}$ is smaller than ${\cal L_0}$ generate a number ($U$) in range [0,1] from a uniform distribution  8) If $U$ is smaller than the difference between the two likelihoods (${\cal L_1}$ - ${\cal L_0}$), accept $a_1$ as the new $a_0$, append it at the end of $C$ and go to step 2.  9) If $U$ is bigger than the difference between the two likelihoods (${\cal L_1}$ - ${\cal L_0}$), append the $a_0$ at the end of $C$, keep using the same $a_0$, go to step 2.  10) End of Repeat.  11) Remove some elements from the start of $C$ (burn-in phase).  12) Now take the average of the values in $C$. This average is the estimated $a$.  Now I have some questions regarding the above steps:   How do I construct the likelihood function for $f(x)=ax$ but also for any arbitrary function? Is this a correct implementation of Metropolis-Hastings algorithm? How the selection of the random number generation method at Step 7 can change the results? How is this algorithm going to change if I have multiple model parameters? For example, if I had the model $f(x)=ax+b$.   Notes/Credits: The main structure of the algorithm described above is based on the code from an MPIA Python Workshop. What does a "Legacy USB Mouse" option in a BIOS mean?  Yes legacy usually means older revision, or obsolete. So legacy usb mouse means support for "old style usb mouses".     That's not really answering the question. As Mr. Tamm, my high school electricity teacher, liked to say, "Yes, and oranges taste orangey."   What does legacy USB mouse support mean?   What is a legacy USB mouse?  There's an "old style" USB mouse?  What changed between what version of usb and what version of usb that made mice incompatible? Why was the change made?  When was the change made? Who made the change? What was the virtue of the "new USB mouses" over the "old USB mouses"?   Put it another way:     What is the BIOS doing when "Legacy USB Mouse" option is enabled?   What is the BIOS doing when "Legacy USB Mouse" option is disabled?   See also Is there an off/disable control anywhere? I realize that the INFO button toggles it off, but it comes right back on as soon as you tap the shutter, and that it a problem in low light situations where the LCD keeps blasting me in the eye. Reputation in summary tab shows as blank instead of 0, if net result is zero If object A is moving at velocity $v$ (normalized so that $c=1$) relative to a ground observer emits object B at velocity $w$ relative to A, the velocity of B relative to the ground observer is  $$ v \oplus w = \frac{v+w}{1+vw} $$  As expected, $v \oplus 1 = 1$,  as "nothing can go faster than light".  Similarly, $v \oplus -1 = -1$. (same thing in the other direction)  But what if object A is moving at the speed of light and emits object B at the speed of light in the exact opposite direction? In other words, what is the value of $$1 \oplus -1?$$ Putting the values in the formula yields the indeterminate form $\frac{0}{0}$. This invites making sense of things by taking a limit, but $$ \lim_{(v,w)\to (1,-1)} \frac{v+w}{1+vw}$$ is not well-defined, because the limit depends on the path taken.  So what would the ground observer see? Is this even a meaningful question?  Edit: I understand $1 \oplus -1$ doesn't make sense mathematically (thought I made it clear above!), I'm asking what would happen physically. I'm getting the sense that my fears were correct, it's physically a nonsensical situation. Deciding what answer to accept: the correct one-liner or the later elaborate one with code? Lately I've been getting some good answers to some specific questions. However, I've been hesitating on what answer I should accept.  Do I go with the short quick answer or do I accept a later answer where someone has elaborated on the previous answer with some code?  It's not really important but it kind of bugs me to not accept an answer when people have taken the trouble of replying.  Here's two examples: Is there any example that for cardinal numbers $\kappa &lt; \lambda$, we have $2^\kappa = 2^\lambda$?  My guess is that it only depends on whether GCH holds. Is it true? Why couldn't Bond have backup at the battle of Skyfall? I think I am missing a key plot point here.   Bond had to fake-kidnap M and bring her to Skyfall in order to lure Silva there for one last stand. Why did this have to be a secret from his colleagues at MI6? Why couldn't there have been other agents there to help defend the house and take down Silva?  Also, why did he have to have Q lay a fake trail to get Silva to follow him to Skyfall when he was actually going there, and he wanted Silva to follow him? My "Meta" rep on Photo.SE differs from my main rep Where does the energy for osmosis come from? A thought experiment:     A U-shaped tube with semi-permeable membrane at the base. The tube is completely thermally isolated from its surroundings. The liquid (solvent) is at some temperature $T$. When solute is added to one side of the tube the fluid level on that side rises until the increased pressure due to gravity ($\rho gh$) equals the osmotic pressure.   If I understand correctly, osmotic pressure could be used to do work (i.e. lift a weight as the solvent level in one side of U-shaped tube increases). Where does this energy come from? I guess it must be from the internal energy of the fluid.  At this point has the temperature of the fluid decreased?? If so, has it decreased by exactly the work done against gravity divided by the specific heat of the fluid? What does the quantitative treatment of this problem look like? The question is to show that for any $x,y \in\Re^n$  $||x+y||\cdot||x-y||\le||x||^2+||y||^2$  with equality if and only if $\langle x,y \rangle= 0.$  I have that $||x+y||^2=||x||^2+2 \langle x,y \rangle+||y||^2$ , which helps me prove equality but I'm lost on how to show this and show that the inequality part. Anime about a boy and a girl who ride ostrich-like creatures through a valley I can not remember much but I have had this film in my head for years I searched everything I can remember on every website I can think of.  I don't remember the exact story but I'll list everything about it can   Apocalyptic future possibly Two main characters: a boy and a girl riding ostrich-like creatures through a valley An old blind lady that senses things Crashed spaceship looking like a submarine Giant creatures looks like yaks with blue eyes, whose eyes turn red when something is about to happen The old lady says "Some say they went to the stars" while looking at the spaceship The two main characters do something and their eyes turn blue in the distance and the village cheers   This film is at least ten years old and I would appreciate it so much if you knew anything about it, or could help me find the title. Why did It not do this to The Losers? Why are there non-visible objects in my final render? How to measure capacitance without LC meter? I would like to use capacitors salvaged from old electronics, but I currently don't have an LC meter.    What would be some really simple circuits, that would convert capacitance to something I can measure (voltage, current, frequency)? Is it possible to navigate to a Visualforce page from an LWC with parameters and community compatible The LWC framework includes the NavigationMixin.Navigate functionality that can be used to obtain URLs or perform navigation to generated URLs. This supports a number of different page types, as covered in the . What I can't find is how to create a URL to a specific Visualforce page, with parameters and compatible with communities. Using the &quot;web page&quot; navigation type seems wrong to me, since the URL for a Visualforce page varies based on whether access is direct or within a community. Do you know what the correct approach is for navigating from a flexipage via an LWC component to a Visualforce page in the same org, passing the page some parameters? Given any uncountable subset $S$ of the unit interval. Then $S$ clearly has an accumulation point and indeed uncountably many (which might also be a nice exercise). So my question is: Is there an accumulation point, that again lies in $S$? How do I exactly find the zero divisors and units of a ring in the form of $\mathbb{Z}_n \times \mathbb{Z}_m$? For instance, how does one calculate zero divisors and units of $\mathbb{Z}_6 \times \mathbb{Z}_2$? I understand that a zero divisor is a non-zero element of $R$ such that $rs = 0$ for some other non-zero element $s$ in $R$. But I don't really quite understand how to apply that to a ring of the form $\mathbb{Z}_n \times \mathbb{Z}_m$. Also, the units of $\mathbb{Z}_6 \times \mathbb{Z}_2$ would be elements that send some element $s$ in $\mathbb{Z}_n \times \mathbb{Z}_m$ to the multiplicative identity, right? (which would be 1)? How do I apply these concepts to "product" rings? Thank you.