The known acceptable range for (for example) an ethernet networking cable before any kind of noticable drop in speed or latency appears is a whole 100 meters for standard class cat5e, cat6 and cat6e cables, and there are variants that can extend this range up to 500 meters and this is for 100mbps, 1gbps and 10gbps speeds depending on what cable you're using...  And I don't know the specs for HDMI, but I'm having absolutely no noticable problems with a 10 meter hdmi cable, and judging by what I saw on the market this should go on for quite some distance beyond this (at least 20m).  Meanwhile over in USB land the maximum distance extension cable I've seen was 3 meters, any longer and it's generally deemed unreliable unless you have a repeater on your cable, and I've actually tested this and if I am extending a USB cable beyond the recommended parameters I start seeing problems (even over 5m with a repeater I have problems) hell, I have a 5 meter cable with no repeater, and it worked for a while shortly after I got it, but then when I tried to connect devices to it after only short use it simply stopped working (windows said unknown/unrecognized device or something like that) and this is only for up to a maximum of 5gbit transfer rates, much lower than what you could acheive with ethernet.  And somehow despite all these problems, it seems like it has no issue with USB Hubs for connecting multiple devices to the same port (as long as they all get powered independently...) but admittedly this too starts degrading if you are using extension cables...  Why does USB have these problems with cable lengths; and only these VERY SHORT lengths when everything else seems to be able to extend pretty much across an entire building without any noticable issues?  And is there anything that can be done to bypass these limitations?
I want to use a very long USB cable to place a webcam in a different room. The maximum length that I found was five meters.   What is the maximum length of a USB cable? What is the maximum length achievable when using USB extension cables?
I am trying to prove  $$\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$$  This problem is a classic, but I seem to be missing one step or the understanding of two steps which I will outline below.   $$I_n := \int\cos^n x \ dx = \int\cos^{n-1} x \cos x \ dx \tag{1}$$  First question: why rewrite the original instead of immediately integrating by parts of $\int \cos^n x \ dx$?  Integrate by parts with $$u = \cos^{n-1} x, dv = \cos x \ dx \implies du = (n-1)\cos^{n-2} x \cdot -\sin x, v = \sin x$$  which leads to  $$I_n = \sin x \ \cos^{n-1} x +\int\sin^2 x (n-1) \ \cos^{n-2} x \ dx \tag{2}$$  Since $(n-1)$ is a constant, we can throw it out front of the integral:  $$I_n = \sin x \ \cos^{n-1} x +(n-1)\int\sin^2 x \ \cos^{n-2} x \ dx\tag{3}$$  I can transform the integral a bit because $\sin^2 x + cos^2 x = 1 \implies \sin^2 x = 1-\cos^2 x$  $$I_n = \sin x \ \cos^{n-1} x + (n-1)\int(1-\cos^2 x) \ \cos^{n-2} x \ dx \tag{4}$$  According to Wikipedia as noted , this simplifies to:  $$I_n = \sin x \ \cos^{n-1} x + (n-1) \int \cos^{n-2} x \ dx - (n-1)\int(\cos^n x) \ dx \tag{5}$$  Question 2: How did they simplify the integral of $\int(1-\cos^2 x) \ dx$ to $\int(\cos^n x) \ dx$?  Assuming knowledge of equation 5, I see how to rewrite it as  $$I_n = \sin x \ \cos^{n-1} x + (n-1) I_{n-2} x  - (n-1) I_{n} \tag{6}$$  and solve for $I_n$. I had tried exploiting the fact that  $$\cos^2 x = \frac{1}{2} \cos(2x) + \frac{1}{2} $$  and trying to deal with $\int 1 \ dx - \int \frac{1}{2} \cos (2x) + \frac{1}{2} \ dx$  which left me with $\frac{x}{2} - \frac{1}{4} \sin(2x)$ after integrating those pieces. Putting it all together I have:  $$I_n = \sin x \ \cos^{n-1} x + (n-1) I_{n-2} x \left(-(n-1) (\frac{x}{2} - \frac{1}{4} \sin 2x) \right) \tag{7}$$  but I'm unsure how to write the last few terms as an expression of $I_{something}$ to get it to match the usual reduction formula of   $$\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$$ Avoid footnotes from appearing before mark Can Ubuntu use all my 4GB of RAM? Regex exec only returning first match What is the difference between cascade & inverse in hibernate, what are they used for? This is what happens when I launch the terminal  There was an error creating a child process  So I have been using the Ubuntu terminal on my Raspberry Pi 4 model B 8 GB with Ubuntu for a couple moths when Ubuntu 20.10 came out for the Pi. But now I have a problem where I launch the terminal and it gives that message. I have gone for alternatives and tried to fix it but all the solutions were outdated or they were on a different distro. Also the Ubuntu commands did not fix the issue because it only triggers upon a certain command. wanted to get end date for Current Fiscal Quarter in apex any ideas, how to get this information. Order of operations using Object Initializer Syntax Does the order in which I set properties using the object initializer syntax get executed in the exact same order?  For instance if I do this:  var s = new Person { FirstName = "Micah",                      LastName = "Martin",                      IsLoaded = true                    }   will each property get set in the same order? I just signed up, a day or two ago, and I've received a notification that there's something in my inbox.  Unfortunately, my email client isn't able to show me a link to my Inbox, or the message, because of the format of the email.  I normally use the email address that I gave when I signed up, just with MailWasher, which is intended to safely preview emails, as plain text, to decide whether or not to open them in a normal HTML email client.  I would be nice if that email contained a link to my inbox, which could be shown in MailWasher.  In order to collect that email in a normal email client, I'd also have to collect or delete all the other emails sent to that address.  I'm not entirely clear what's been done in that email to stop MailWasher seeing the link.  I can see a link to leave feedback, and one to unsubscribe, but not one to check my inbox.  Sure, I can live without one, but those emails would be a lot more useful to me if I could just click a link to go to my Inbox. Is it possible to modify the title of the message box the confirm() function opens in JavaScript?   I could create a modal popup box, but I would like to do this as minimalistic as possible. I would like to do something like this:  confirm("This is the content of the message box", "Modified title");   The default title in Internet Explorer is "Windows Internet Explorer" and in Firefox it's "[JavaScript-program]." Not very informative. Though I can understand from a browser security stand point that you shouldn't be able to do this. I understand the energy and mass can change back and forth according to Einstein. It is fluid; it can go from one to the other. So, what keeps mass from just turning into energy? Is there some force holding a subatomic particle together? What keeps mass in it's state? I hope this is not a silly question but I am clueless. Thanks When I launch nautilus from terminal the following error is printed and nautilus doesn't open up.  (nautilus:25179): GLib-GIO-CRITICAL **: g_dbus_interface_skeleton_unexport: assertion 'interface_-&gt;priv-&gt;connections != NULL' failed  (nautilus:25179): GLib-GIO-CRITICAL **: g_dbus_interface_skeleton_unexport: assertion 'interface_-&gt;priv-&gt;connections != NULL' failed Could not register the application: Timeout was reached  (nautilus:25179): Gtk-CRITICAL **: gtk_icon_theme_get_for_screen: assertion 'GDK_IS_SCREEN (screen)' failed  (nautilus:25179): GLib-GObject-WARNING **: invalid (NULL) pointer instance  (nautilus:25179): GLib-GObject-CRITICAL **: g_signal_connect_object: assertion 'G_TYPE_CHECK_INSTANCE (instance)' failed   What could be the problem?? Make links posted by users open in a new window Most of the time when I follow a user's link, I just want to see their site, look it over, then return to answer their question (not leave SO completely and not come back). Could we make the links users post open in a new window please?  EDIT: I originally wanted this implemented because it's what some other sites did, but I now definitely prefer SO's way. If I want to open a new tab, I middle-click/control-click (or whatever); if not, I just click. I'm in control of what happens. Would never want to go back now. I've been exploring implementing Angular 2.0 in Salesforce custom pages and found that if I set showHeader=true in the VF page, I run into some issues.   It looks like including the header loads "sfdc/main.js", which defines the browser's Map() function without checking if one already exists. This removes some needed prototype functions that are native to ES5/6 such as "set" (needed in Angular 2) and "delete" (needed for an ES6-shim).   I've found that even including an ES5 shim before the ES6 one in attempts to restore some of these prototype methods does not work. If I set showHeader=false, "sfdc/main.js" is not loaded and everything works fine.   Is there a recommended way around this, or is it possible to update "sfdc/main.js" to conditionally define Map() if one does not already exist? It looks like it leaves out quite a few native functions supported in modern browsers.  Thanks Possible Duplicates:             I heard a professor of mathematics explaining that the following conslusions are true.   $\textrm{False}\Rightarrow \textrm{True}$ $\textrm{False}\Rightarrow \textrm{False}$   Could someone please explain to me why both conclusions are true? I am especially interested in the first case. I'd like to hear two explanations if possible: One for mathematicians and one for non-mathematicians. :-) I have seen a "Universal Laptop Adapter" sold in a computer shop before. I am thinking of getting one for a laptop. Is it safe to use? What differences are there between one of these and a typical laptop adapter? Why should I "watch" a nearby Pokémon? Reason behind choosing the compensating resistor for input bias current in op amps I found the following explanation online on determining the values used for the compensating resistor:  In either case, the compensating resistor value is determined by calculating the parallel resistance value of R1 and R2. Why is the value equal to the parallel equivalent of R1 and R2? When using the Superposition Theorem to figure how much voltage drop will be produced by the inverting (-) input's bias current, we treat the bias current as though it were coming from a current source inside the op-amp and short-circuit all voltage sources (Vin and Vout). This gives two parallel paths for bias current (through R1 and through R2, both to ground). We want to duplicate the bias current's effect on the noninverting (+) input, so the resistor value we choose to insert in series with that input needs to be equal to R1 in parallel with R2.  Although it is a 'crisp' and simple explanation, I don't understand how it is true, since the current flowing through R2 won't directly go to the ground, but will probably enter the op-amp at the output (op-amp as a current sink). Also, it might go through a few more resistances before going to the ground, since Vout isn't directly connected to the ground. What will the universe look like for anyone falling into a black hole? I've heard that, from the perspective of an external observer, something falling into a black hole will eventually look "frozen": light waves will move to the infrared and further into lower frequencies.  But what about that unfortunate individual falling into such a black hole? What will he/she see when looking back at our universe?   For instance, I wonder, will it be the ultimate end of the Universe? Can someone who just crossed events horizon of the black hole still receive messages from our side? Everytime, I read about blackbody, I always get confused at the point where it is written     Under thermal equilibrium conditions , the blackbody radiation depends only on temperature. ..... . At equilibrium , electromagnetic waves bounce around with the walls of the cavity.   What is this equilibrium all about? Why is it important here?? If there is thermal equilibrium, why does the body radiate?? Please help.