I try to prove that these four statements are equivalent:   $\Diamond$ There are $A_\alpha\subseteq \alpha\times\alpha$ for $\alpha&lt;\omega_1$ s.t. for all $A\subset \omega_1\times\omega_1$, $\{\alpha&lt;\omega_1:A\cap\alpha\times\alpha=A_\alpha\}$ is stationary. There are $f_\alpha:\alpha\to\alpha$ $\alpha&lt;\omega_1$ s.t. for each $f:\omega_1\to\omega_1$, $\{\alpha:f\upharpoonright\alpha=f_\alpha\}$ is stationary. There are $f_\alpha:\alpha\to\alpha$ $\alpha&lt;\omega_1$ s.t. for each $f:\omega_1\to\omega_1$, there is $\alpha&gt;0$ such that $f\upharpoonright \alpha=f_\alpha$.   I prove that 1, 2 and 3 are equivalent and 3 implies 4, but I don't know how to prove the equivalence of 4 and 1. I try to prove that 4 implies 3 or $\lnot 2$ implies $\lnot 3$ however I have a struck. Thanks for any hints.
I'm working on exercises from Kunen and I'm stuck. I must proof that the following are equivalent:   There exists a sequence $\langle A_\alpha:\alpha&lt;\omega_1\rangle$ such that $\forall\alpha(A_\alpha\subset\alpha)$ and for all $A \subset \omega_1$ the set $\{\alpha \in \omega_1:A\cap\alpha=A_\alpha\}$ is stationary. ($\diamondsuit$-principle) There exists a sequence $\langle A_\alpha:\alpha&lt;\omega_1\rangle$ such that $\forall\alpha(A_\alpha\subset\alpha\times\alpha)$ and for all $A \subset \omega_1\times\omega_1$ the set $\{\alpha \in \omega_1:A\cap\alpha\times\alpha=A_\alpha\}$ is stationary. There exists a sequence $\langle f_\alpha:\alpha&lt;\omega_1\rangle$ such that $\forall\alpha(f_\alpha:\alpha\rightarrow\alpha)$ and for all $f : \omega_1\rightarrow\omega_1$ $\exists\alpha(f|\alpha=f_\alpha\wedge\alpha&gt;0)$ There exists a sequence $\langle f_\alpha:\alpha&lt;\omega_1\rangle$ such that $\forall\alpha(f_\alpha:\alpha\rightarrow\alpha)$ and for all $f : \omega_1\rightarrow\omega_1$ the set $\{\alpha \in \omega_1:f|\alpha=f_\alpha\}$ is stationary.   I have proved that $1\implies 2\implies 4\implies 3$ and that $4\implies 1,$ but I'm having problems on proving that $3\implies$ "something", because it only talks about one $\alpha$. Can someone help me?
As a  developer I'm used to running through constructors:    class Test {     public Test() {         DoSomething();     }      public Test(int count) : this() {         DoSomethingWithCount(count);     }      public Test(int count, string name) : this(count) {         DoSomethingWithName(name);     } }   Is there a way to do this in C++?  I tried calling the Class name and using the 'this' keyword, but both fail. Asymptotic expansion of $u_{n + 1} = \frac12 \arctan(u_n)$ Meaning of hash symbol in pin name I'm reading the datasheet of a PSRAM (available ). In the top-right hand corner all the pins are listed. Some of the pin names end with a hash, e.g. LB#, OE#.  What does the # notation mean for pins? How do I erase an iPhone? I had an iPhone on loan from a friend to try it out and now I need to give it back. How do I make sure that everything is erased from it? SMS, Calendar, Phone Logs, contacts? How to change programmatically exposed filters in views I have a page view with exposed filters. One of the filters is term vocabulary, and that's dropdown list, user can select only one category (term).   I want to set programmatically one term from dropdown list when I came from specific URL. I tried with views_query_alter where $category is tid.    $set_exposed_input  = array();  $set_exposed_input['field_category_tid'] = $category;   if ($set_exposed_input) {     $view-&gt;set_exposed_input($set_exposed_input);  } I am having a problem with permissions on a Linux server. I am used to BSD. When a directory is owned by a group the user who owns it isn't in such as www-data, files created in it will be owned by that group. This is important because I want files to be readable by the webserver (which I will not run as root) but so a user can still put new files in the directory. I can't put the users in www-data because then they can read every other users websites.   I want the webserver to read all websites, I want users to be able to change their own.  The permissions are set like this on the folders at the moment....  drwxr-x--- 3 john www-data 4096 Feb 17 21:27 john   It is standard behavior on BSD for permissions to work this way. How do I get Linux to do this? How to set a JavaScript breakpoint from code in Chrome? I want to assign the result of an expression (i.e., the output from a command) to a variable and then manipulate it – for example, concatenate it with a string, then echo it.  Here's what I've got: #!/bin/bash cd ~/Desktop; thefile= ls -t -U | grep -m 1 &quot;Screen Shot&quot;; echo &quot;Most recent screenshot is: &quot;$thefile;  But that outputs: Screen Shot 2011-07-03 at 1.55.43 PM.png Most recent screenshot is:   So, it looks like that isn't getting assigned to $thefile, and is being printed as it's executed. What am I missing? My PC (Lenovo Ideapad z580) is overheating while running Ubuntu GNOME 16.04 LTS.  $ uname -a Linux Mahadev 4.4.0-72-generic #93-Ubuntu SMP Fri Mar 31 14:07:41 UTC 2017 x86_64 x86_64 x86_64 GNU/Linux  $ lspci -nnk|grep -A2 VGA 00:02.0 VGA compatible controller [0300]: Intel Corporation 3rd Gen Core processor Graphics Controller [8086:0166] (rev 09)   Subsystem: Lenovo 3rd Gen Core processor Graphics Controller [17aa:3977]   Kernel driver in use: i915   $ sensors acpitz-virtual-0 Adapter: Virtual device temp1:        +52.0°C  (crit = +104.0°C)  coretemp-isa-0000 Adapter: ISA adapter Physical id 0:  +54.0°C  (high = +72.0°C, crit = +90.0°C) Core 0:         +54.0°C  (high = +72.0°C, crit = +90.0°C) Core 1:         +54.0°C  (high = +72.0°C, crit = +90.0°C)  update vivek@Mahadev:~$ stress -t 300 -c $(getconf _NPROCESSORS_ONLN) &amp;&amp;  sensors stress: info: [4408] dispatching hogs: 4 cpu, 0 io, 0 vm, 0 hdd stress: info: [4408] successful run completed in 300s acpitz-virtual-0 Adapter: Virtual device temp1:        +73.0°C  (crit = +104.0°C)  coretemp-isa-0000 Adapter: ISA adapter Physical id 0:  +72.0°C  (high = +72.0°C, crit = +90.0°C) Core 0:         +69.0°C  (high = +72.0°C, crit = +90.0°C) Core 1:         +72.0°C  (high = +72.0°C, crit = +90.0°C)  update     vivek@Mahadev:~$ sudo lshw -C cpu     [sudo] password for vivek:    *-cpu                             description: CPU         product: Intel(R) Core(TM) i3-3110M CPU @ 2.40GHz         vendor: Intel Corp.        physical id: 0        bus info: cpu@0        version: Intel(R) Core(TM) i3-3110M CPU @ 2.40GHz        serial: To Be Filled By O.E.M.        slot: CPU Socket - U3E1        size: 1302MHz        capacity: 2400MHz        width: 64 bits        clock: 100MHz        capabilities: x86-64 fpu fpu_exception wp vme de pse tsc msr  pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush dts acpi mmx  fxsr sse sse2 ss ht tm pbe syscall nx rdtscp constant_tsc arch_perfmon  pebs bts rep_good nopl xtopology nonstop_tsc aperfmperf eagerfpu pni    pclmulqdq dtes64 monitor ds_cpl vmx est tm2 ssse3 cx16 xtpr pdcm pcid    sse4_1 sse4_2 x2apic popcnt tsc_deadline_timer xsave avx f16c lahf_lm    epb tpr_shadow vnmi flexpriority ept vpid fsgsbase smep erms xsaveopt     dtherm arat pln pts cpufreq           configuration: cores=2 enabledcores=2 threads=4  Update This time I edited the sensors , htop command result in image. It's getting overheated, my laptop is getting hang cause of overheating, please watch and reply. Thanks. Wasted screen space due to multiple horizontal bars at the top when an application is open in GNOME desktop Just moved from Unity to GNOME with the update to Ubuntu 17.10. There is a HUGE amount of wasted space at the top of the screen, when an application is open.  Evolution and Firefox both have four "bars" with 90% of each empty. Unity, to its credit was able to blend / mix these horizontal bars so there was only one / two.      Four is way too much. If this is GNOME I may carry on with Unity for the next six months.  Can anybody suggest a tweak/change/fix for this? Undefined variable problem with PHP function Given $f_{n} : [a,b]\rightarrow \mathbb R$ is uniformly Lipschitz and $f_{n}\rightarrow f$, how can I show that $f$ is Lipschitz?  $f_{n}$ is uniformly Lipschitz $\leftrightarrow \exists K &gt; 0$ such that for all $n$ and for every $x,y\in [a,b]$ we have $|f_{n}(x)-f_{n}(y)| &lt; K|x-y|$.  $f_{n}\rightarrow f \leftrightarrow \forall~\epsilon&gt;0~~\exists~N$ such that $\forall~n&gt;N$ we have $\mid f_{n}(x) - f(x)\mid&lt;\epsilon~~~(=\mid x-y\mid?)$  Then we have  \begin{align} |f(x)-f(y)| &amp;\leq |f(x)- f_{n}(x)| + |f_{n}(y)-f(y)| + |f_{n}(x)-f_{n}(y)| \\ &amp;\leq |x-y| + |x-y| + K|x-y| \\ &amp;\leq (K + 2) |x - y| \end{align}  which shows that $f$ is Lipschitz. Can anybody tell me if this makes sense? Sorry if layout is poor. Stop asking me if Community auto-flags were helpful How to nest code within a list using Markdown Oftentimes, I want to include code samples in a list, for example:   Item One Item Two, for example: private bool ItemTwo() {     return this.IFeelLucky; } Item Three   The problem is that there isn't a good and intuitive way to do this using . It's doable, but you have to do some formatting black-magic to get it that way, which is just not "nice" to deal with.  Source that produces this:  1. Item One 2. Item Two, for example: &lt;br/>&lt;br/>&lt;pre>&lt;code>private bool ItemTwo()     {         return this.IFeelLucky;     }&lt;/code>&lt;/pre> 3. Item Three    I would love to just be able to do it this way:  1. Item One 2. Item Two, for example:      private bool Test()     {         return this.IFeelLucky;     }  3. Item Three  Which results in, well, this:   Item One Item Two, for example:  private bool Test() {     return this.IFeelLucky; } Item Three It seems that panel shadow is somehow hardcoded in KDE 4 and devs do not have any plans to change it and/or make it customizable. So, what I'm actually asking is some kind of hack (if there is any) which will remove the shadow of the panel in KDE 4.x completely (but without need to disable all desktop effects in the process).  Possible application: When a user wants a completely transparent panel, no top/bottom borders at all, the shadow is completely out of place and destroys the experience. I am an Indian citizen. I am flying from Toronto to New Delhi. I have two transits: one at Lisbon airport and the second one at Madrid airport.  While the Consulate of Portugal has assured me that I do not need an airport transit visa during my 2 hours stay in Lisbon airport, I did not get a definitive answer from the consulate of Spain in Toronto.   At Madrid airport, I need to make a transfer from Terminal 2 to Terminal 1.   Does anyone here know whether I need to pass through immigration while making this transition in Madrid and if so, do I need a Schengen visa for that? I know that $\operatorname{Spec} \mathbb{C}[x]$ can be identified with the set $\mathbb{C}\cup *$, where $*$ is a generic point via the correspondence  $$ \prod_{i}(x-a_i) \leftrightarrow \{a_i\}_i , \ \ \ \ (0)\leftrightarrow *. $$ This correspondence holds for any $\operatorname{Spec} k[x]$ with algebraic closed field $k$.   When $k$ is not algebraic closed, how should one understand $\operatorname{Spec} k[x]$? Best way to track onchange as-you-type in input type="text"? In my experience, input type="text" onchange event usually occurs only after you leave (blur) the control.  Is there a way to force browser to trigger onchange every time textfield content changes? If not, what is the most elegant way to track this “manually”?  Using onkey* events is not reliable, since you can right-click the field and choose Paste, and this will change the field without any keyboard input.  Is setTimeout the only way?.. Ugly :-) Is there an easy way to calculate the total cost of tolls for a trip within the United States? Finding $\lim \limits_{x \to 0} \frac{1 - \cos x}{x}$, given $\lim \limits_{x \to 0} \frac{\sin x}{x} = 1$