Implode and Explode Multi dimensional arrays
Return single column from a multi-dimensional array
I don't have complex analysis at my beck and call, and I only have a low level of knowledge in topology, but I need to prove that this metric space (for any real $r$ and $R$ with $r &lt; R$)$$ X = \{ (x, y) \in \mathbb{R}^2 \ | \ r \leq x^2 + y^2 \leq R \}$$  with the Manhattan metric $d((x_1, y_1), (x_2, y_2)) = |x_1-x_2| + |y_1-y_2|$ is not simply connected.  I've already prooven that it's path connected, and now I need to show there are some points $P$ and $Q$ with two paths between them such that one cannot be continuously 'morphed' into the other.  $ \ $  What I have so far is as follows:  I take $P = (0, r)$ and $Q = (0, -r)$, with $f_0$ being a path from $P$ to $Q$ going clockwise around the circle radius $r$ and $f_1$ being much the same but going counterclockwise.  Now I assume there is a function $g : [0, 1]^2 \to X$ such that:   $g(s, 0) = f_0 (s)$ $g(s, 1) = f_1 (s)$ $g(0, t) = P$ $g(1, t) = Q$   To get the final result, I need to show that this function cannot be continuous, but for the life of me I cannot.  For some context, these are the topics which have been visited during the course, roughly in order of recentness.   Multiple connectedness Simple connectedness Pathwise connectedness Interior points Boundary points Open sets Compactness Complete metric spaces Bounded metric spaces Totally bounded metric spaces Closed sets Closure of a metric space Limit points Cauchy sequences Convergence Continuity Metric space quivalence Metric equivalence   Edit: I might have an argument that works, though it's far from rigourous. We can shrink $R$ to be as close to $r$ as we want, so we can essentially constrain the annulus down to a circle and thus force any path from the left side of the circle to the right side to go through $P$ or $Q$.  So holding $s \in (0, 1)$ constant and varying $t$ must produce a path through $P$ or $Q$ for any $s$. If for some $s$ it passes through $P$ and for some other $s$ it passes through $Q$, then there must be $s_0$ such that $\forall \epsilon &gt;0  \ \exists \delta \leq \epsilon$ st. $s_0$ produces a path through $P$ and $s_0 + \delta$ produces a path through $Q$.  Now we can consider the path $g(s, \frac{1}{2})$, and note that it must have a discontinuity at $s_0$.  Now consider the case where all $s$ produce paths through only one of $P$ or $Q$. WOLOG: $P$. Now, at $t = \frac{1}{2}$, $s$ arbitrarily close to $1$ are mapped away from $Q$, but $1$ is always mapped to $Q$ by definition, so $g(s, \frac{1}{2})$ has a discontinuity at $s=1$. Therefore $g(s, t)$ is not continuous.  This argument is definitely iffy to me, if no one has their own argument (using sufficiently low level concepts), then criticism on the above would be appreciated. If a linear operator has an adjoint operator, it is bounded How to combine measurement error with statistic error We have to measure a period of an oscillation. We are to take the time it takes for 50 oscillations multiple times.  I know that I will have a $\Delta t = 0.1 \, \mathrm s$ because of my reaction time. If I now measure, say 40, 41 and 39 seconds in three runs, I will also have standard deviation of 1.  What is the total error then? Do I add them up, like so?  $$\sqrt{1^2 + 0.1^2}$$  Or is it just the 1 and I discard the (systematic?) error of my reaction time?  I wonder if I measure a huge number of times, the standard deviation should become tiny compared to my reaction time. Is the lower bound 0 or is it my reaction time with 0.1? Even my main monitor is shown as unknown. I have tried setting up dual monitors using Nvidia settings with no joy however it does recognize that I have 2 monitors.  This was not an issue in 11.04. My problem with the Nvidia settings appears to be that my 2 monitors require different resolutions but even when I set them to a resolution that they both can handle, it is then only appearing in cloned mode. I really hope someone can help or I will have to roll back to 11.04.  Why won't Ubuntu 11.10 recognize that I have 2 Monitors?   Here is the contents of my xorg.conf:  # nvidia-settings: X configuration file generated by nvidia-settings # nvidia-settings:  version 280.13  (buildd@rothera)  Thu Aug 11 17:28:49 UTC 2011  Section "ServerLayout"     Identifier     "Layout0"     Screen      0  "Screen0" 0 0     InputDevice    "Keyboard0" "CoreKeyboard"     InputDevice    "Mouse0" "CorePointer"     Option         "Xinerama" "0" EndSection  Section "Files" EndSection  Section "InputDevice"     # generated from default     Identifier     "Mouse0"     Driver         "mouse"     Option         "Protocol" "auto"     Option         "Device" "/dev/psaux"     Option         "Emulate3Buttons" "no"     Option         "ZAxisMapping" "4 5" EndSection  Section "InputDevice"     # generated from default     Identifier     "Keyboard0"     Driver         "kbd" EndSection  Section "Monitor"     # HorizSync source: xconfig, VertRefresh source: xconfig     Identifier     "Monitor0"     VendorName     "Unknown"     ModelName      "CRT-0"     HorizSync       30.0 - 83.0     VertRefresh     50.0 - 76.0     Option         "DPMS" EndSection  Section "Device"     Identifier     "Device0"     Driver         "nvidia"     VendorName     "NVIDIA Corporation"     BoardName      "GeForce 7025 / nForce 630a" EndSection  Section "Screen"     Identifier     "Screen0"     Device         "Device0"     Monitor        "Monitor0"     DefaultDepth    24     Option         "TwinView" "1"     Option         "TwinViewXineramaInfoOrder" "CRT-0"     Option         "metamodes" "CRT: 1280x1024 +1024+0, DFP: 1024x768 +0+0"     SubSection     "Display"         Depth       24     EndSubSection EndSection"   ok I have tried disper -e. Problem is that Nvidia settings appear to be treating both monitors as the same one. so changing the settings to one has no effect, but change the other and it affects both screens identically.  Results of disper -l as follows:      carter@carter-desktop:~$ disper -l display DFP-0: HP vs17  resolutions: 320x240, 400x300, 416x312, 512x384, 576x432, 680x384, 640x480, 720x450, 640x512, 800x512, 840x525, 800x600, 960x540, 832x624, 960x600, 896x672, 1024x768, 1152x864, 1280x960, 1280x1024 display CRT-0: CRT-0  resolutions: 320x240, 400x300, 416x312, 512x384, 576x432, 680x384, 640x480, 720x450, 640x512, 800x512, 840x525, 800x600, 960x540, 832x624, 960x600, 896x672, 1024x768, 1152x864, 1360x768, 1280x960, 1440x900, 1280x1024, 1400x1050, 1600x1024, 1680x1050, 1600x1200, 1920x1080, 1920x1200, 1792x1344 carter@carter-desktop:~$ How do I make the Super key open the activities overview in Ubuntu 20.04 LTS? How to invoke a system call via syscall or sysenter in inline assembly? I've got a Windows 10 Computer (home edition), which has 4GB of Ram and an i5 processor. I need to run an old 32 bit accounting programme. The vendor said it won't release a 64 bit version (although a 64bit version exists, it's missing key features from the older version).  The programme installs fine, but when I click on the programme icon, it says the programme won't work on a 64 bit system.   What options do I have?    I've done a fair bit of research, but it seems to run up dead ends.   1) WoW64     Apparently, this is supposed to let you run 32 bits on 64 bit Windows. However, I am not sure how to install it. I think it's for web servers only. However,  says it can work on Windows 7, but doesn't explain how to set it up (or if it will work on Windows 10).  2) XP mode     A 32bit Windows XP emulator. However, it seems to be for Windows 7 only. Also according to this , you need Windows Professional.   3) Virtual Box Machine    Let's you emulate Windows. Seems overkill for my purposes. Plus I would need to buy a 32 bit Windows license. I think it would run slowly with 4GB of Ram.  4) Partition drive into 32bit and 64 bit versions Apparently, you can use the free tool  to split the drive into two and then install separate 32bit and 64 bit versions.   Again, seems overkill and it looks like I would need buy a 32 bit Windows 10 license.   Are these my only options?     Images  Installation    First run (From the error message I guess it's a 16-bit program)   Clicking on programme What does pwd output? Does the command pwd in a shell script output the directory the shell script is in? Consider the sentence      E-books are on the rise, but they haven't suppressed paper books though.    This usage seems to be quite common, but when I learned English  I was taught to use "however" where I now read "though". My questions on this use of "though" are:     Is this a new trend? Is it restricted to American English? Is "though"  considered more colloquial than "however" ?    And on a syntactic level: Do you put a comma before "though"? Why copy constructor not invoked here Use a variable reference "inside" another variable I'm sure it is relatively simple, I just don't know how to do it.  #!/usr/bin/ksh set `iostat` myvar=6   I want to something like echo ${$myvar} which i want interpreted as ${$myvar} -> ${6} -> value How to set up a wi-fi hotspot with an Ubuntu laptop (access point mode)? I've read , and carefully followed the instructions provided. I am sure the wireless card in my laptop supports going into wireless access point mode. But unfortunately, when I use the following command to start hostapd service it throws an error:   $ sudo service hostapd stop  Starting advanced IEEE 802.11 management hostapd                      [fail]    The instructions mentioned above said this could possibly be because my wireless card does not support the g mode, and provided the wiki of hostapd. But what is g mode? I tried to switch to a, or b mode, but it just didn't work.  Can anyone help with this or recommend a tool like connectify in windows ?  ps. I am using Ubuntu 12.04, 32 bit. I have an ADSL wired network, and I want to share it by setting my laptop into a WiFi hotspot with  my windows phone. Distributivity of quantifiers over logical connectives In an extract I am dealing with, I have 2 datetime columns. One column stores the dates and another the times as shown.  How can I query the table to combine these two fields into 1 column of type datetime?  Dates  2009-03-12 00:00:00.000 2009-03-26 00:00:00.000 2009-03-26 00:00:00.000   Times  1899-12-30 12:30:00.000 1899-12-30 10:00:00.000 1899-12-30 10:00:00.000 I know how to rename files in Unix:  $ mv ~/folder/subfolder/file.txt ~/folder/subfolder/file.sh       ^-------this part------^    ^------this part-------^   It takes too long time to repeat ~/folder/subfolder/file twice.  Is there a quicker way? Completion handlers and return values The attribute image.itemtype has an invalid value I am using JSON-LD format for Structured data. In home page @type=website &amp; in details page @type=Article. Logo property of Organization section in details page showing following warning in Google Structured Data Testing Tool But home page is not although in both page value is same.      (The attribute logo.itemtype has an invalid value.)   Home page JSON-LD  &lt;script type="application/ld+json"&gt;     {     "@context": "http://schema.org",     "@type": "WebSite",     "url": "http://kothaboli.com/",     "name": ".....",     "alternateName" : "....",     "description" : "......",     "potentialAction": {         "@type": "SearchAction",         "target": "http://kothaboli.com/search?keywords={search_term_string}",         "query-input": "required name=search_term_string"     },     "sameAs":[         "https://www.facebook.com/kothaboliDotCom/",         "https://twitter.com/kotha_boli",         "https://plus.google.com/+kothaboli"     ],     "publisher" : {         "@type" : "Organization",         "name" : "....",         "alternateName": "...",         "logo": "http://kothaboli.com/media/1018/ka.jpg"         }     } &lt;/script&gt;     Details page JSON-LD      &lt;script type="application/ld+json"&gt;     {     "@context" : "http://schema.org",     "@type" : "Article",     "url": "http://kothaboli.com/interviews/intellectual/arundhati-roy/",     "name" : "... ....",     "headline":  "...--’",     "datePublished" : "7/26/2016 1:10:05 AM",     "dateModified" : "8/12/2016 1:50:54 AM",     "image" : "http://kothaboli.com/media/1088/33333333.jpg",     "articleSection" : "....",     "articleBody" : ".....’ ",     "mainEntityOfPage": "..",     "author" : {         "@type" : "Person",         "name" : "....",         "alternateName": "Arundhati Roy"         },     "publisher" : {         "@type" : "Organization",         "name" : "...",         "alternateName": "Kothaboli",         "logo": "http://kothaboli.com/media/1018/ka.jpg"         }     } &lt;/script&gt; C# vs Java Enum (for those new to C#) basic geometry angle problem Two straight lines, AB and AD, lie in the same vertical plane. AB makes an angle of 25 degrees with the horizontal, and AD makes 32 degrees with the vertical. What is the angle between AB and AD?  The answer is 33 degrees or 93 degrees.   I cannot visualize what this question is asking. It says both AD and AD lie on the same vertical plane, then immediately says AD makes 32 degrees with the vertical. Isn't this a contradiction? How to find collisions in a hash function knowing its coefficent? Say I have a simple hash function where $x$ is a sequence of integers:  $$h(x)=(a_1\cdot x_1+...+ a_n\cdot x_n) \bmod N=\sum_i a_i\cdot x_i \bmod N$$  where $a_1,a_2,a_3,...,a_n$ is the coefficient known to the attacker, size of $x$ can be anything smaller than $N$.   How to find 2 different inputs that hashes to the same value, i.e $0$?  I feel like this has something to do with linear combination / euclid algorithm but I'm not sure how to proceed.