Given any $\beta \in R$, do there exist integers $k,l,m$, and a real number $x$ such that:  $l=4x^{3}m+6x^{2}m^{2}+4xm^{3}-2xm$,  as well as  $x^{4}-x^{2}=k+\beta$ ?
An  can be identified by its  together with isolating intervals with rational bounds for its real and imaginary parts. The degree of an algebraic number is the  of its minimal polynomial. There are known algorithms that allow to easily compute sum and product of algebraic numbers in this representation, raise them to a rational power, extract real and imaginary parts, compare them, or evaluate them numerically to an arbitrary precision.  Is there an efficient algorithm that given an algebraic number $\alpha$ in this representation can decide if $\alpha$ can be represented as a sum or product of algebraic numbers of smaller degree?
The determinant of block triangular matrix as product of determinants of diagonal blocks I am given the following partitioned, upper-triangular matrix:  $$ \begin{bmatrix}   A_1 &amp;*  &amp;*  &amp;*  &amp;*  &amp;*  \\    0&amp; A_2 &amp;*  &amp;*  &amp;*  &amp;*  \\    .&amp;  0&amp; A_3 &amp;*  &amp;*  &amp;*  \\    .&amp;  0&amp;  0 &amp;...  &amp;*  &amp;. \\    .&amp;  0&amp;  0&amp;  0&amp; ... &amp;.  \\    0&amp;  .&amp;  ...&amp;  0&amp;0 &amp; A_m \end{bmatrix} $$  where each of the $A_i$ are block-matrices.  I have to prove that the determinant of this general matrix is: $$ \prod_{n=1}^m \det A_i. $$  We already proved in class that it is true for $ m=2 $ and we can use it in our proof.  Thanks for your answers! How can I get my external IP address in a shell script? Use of inequality $1 - \cos (x) \leq x^2 /2$ In  answer the value of $1 - \cos(x)$ has to be evaluated in order to find its upper limit, if it exists.  In particular, $x = 2 \pi / n$. The answer is related to the length of a side of a regular $n$-gon inscribed into a unit-radius circumference; because the perimeter of the $n$-gon is always less than $2 \pi$, the single side must always be less than $2 \pi / n$.  The inequality  $$1 - \cos (x) \leq \displaystyle \frac{x^2}{2}$$ (1)  is used and the proof is completed with   $$2(1 - \cos(x)) \leq (2 \pi / n)^2$$  $$\sqrt{2(1 - \cos(x))} \leq 2 \pi / n$$  But it is well known that the cosine is a function $f(x) \in [-1;1]$, so $1 - \cos (x) \in [0,2]$. By using this information, we would obtain  $$1 - \cos (x) \leq 2$$ (2)  The proof would provide  $$2(1 - \cos(x)) \leq 4$$  $$\sqrt{2(1 - \cos(x))} \leq 2$$  which is a completely different result.   Why in that case it is preferable to use (1) instead of (2)? How to choose when it is convenient to use (1) and when to use (2) in a proof? The official Checkpoint out command line tool from CheckPoint, for setting up a SSL Network Extender VPN is not longer working from the Linux command line. It is also no longer actively supported by CheckPoint.  However, there is a promising  project, that tries to replicate the Java applet for authentication, that talks with the snx command line utility, called snxconnect.   I was trying to put snxconnect text utility to work in Debian Buster, doing:  sudo pip install snxvpn   and  export PYTHONHTTPSVERIFY=0 snxconnect -H checkpoint.hostname -U USER    However, it was mostly dying either with an HTTP error of:   HTTP/1.1 301 Moved Permanently:   or:  Got HTTP response: HTTP/1.1 302 Found   or:  Unexpected response, try again.   What to do about it?  PS. The EndPoint Security VPN official client is working well both in a Mac High Sierra and Windows 10 Pro. CPU governor changes automatically to "performance" under load I have a Lenovo T570 with Xubuntu 18.04 LTS. My current issue is that my cpu speed stepping is kind of going nuts and does what ever it thinks its best.  Sample: when I run my IDE and cpu load increases during build time, the governor changes automatically. I caught the moment during compiling my project with the following call:  martin@martin-ThinkPad-T570:~$ while cat /sys/devices/system/cpu/cpu0/cpufreq/scaling_governor; do sleep 1; done powersave powersave performance performance   cpufreq-info tells me this:  martin@martin-ThinkPad-T570:~$ cpufreq-info  cpufrequtils 008: cpufreq-info (C) Dominik Brodowski 2004-2009 Report errors and bugs to cpufreq@vger.kernel.org, please. analyzing CPU 0:   driver: intel_pstate   CPUs which run at the same hardware frequency: 0   CPUs which need to have their frequency coordinated by software: 0   maximum transition latency: 4294.55 ms.   hardware limits: 400 MHz - 3.50 GHz   available cpufreq governors: performance, powersave   current policy: frequency should be within 3.50 GHz and 3.50 GHz.                   The governor "performance" may decide which speed to use                   within this range.   current CPU frequency is 3.25 GHz. analyzing CPU 1:   driver: intel_pstate   CPUs which run at the same hardware frequency: 1   CPUs which need to have their frequency coordinated by software: 1   maximum transition latency: 4294.55 ms.   hardware limits: 400 MHz - 3.50 GHz   available cpufreq governors: performance, powersave   current policy: frequency should be within 3.50 GHz and 3.50 GHz.                   The governor "performance" may decide which speed to use                   within this range.   current CPU frequency is 3.36 GHz. analyzing CPU 2:   driver: intel_pstate   CPUs which run at the same hardware frequency: 2   CPUs which need to have their frequency coordinated by software: 2   maximum transition latency: 4294.55 ms.   hardware limits: 400 MHz - 3.50 GHz   available cpufreq governors: performance, powersave   current policy: frequency should be within 3.50 GHz and 3.50 GHz.                   The governor "performance" may decide which speed to use                   within this range.   current CPU frequency is 3.26 GHz. analyzing CPU 3:   driver: intel_pstate   CPUs which run at the same hardware frequency: 3   CPUs which need to have their frequency coordinated by software: 3   maximum transition latency: 4294.55 ms.   hardware limits: 400 MHz - 3.50 GHz   available cpufreq governors: performance, powersave   current policy: frequency should be within 3.50 GHz and 3.50 GHz.                   The governor "performance" may decide which speed to use                   within this range.   current CPU frequency is 3.36 GHz.   Please take a look at     current policy: frequency should be within 3.50 GHz and 3.50 GHz.       The governor "performance" may decide which speed to use   Afterwards to go back to powersave I use my little script:  martin@martin-ThinkPad-T570:~$ cat cpu_freq_save.sh  #!/bin/bash sudo cpufreq-set -g powersave -c 0 --min 400MHz sudo cpufreq-set -g powersave -c 1 --min 400MHz sudo cpufreq-set -g powersave -c 2 --min 400MHz sudo cpufreq-set -g powersave -c 3 --min 400MHz   Which resets the min frequency and the governor but as soon as I compile again with my IDE every time the governor and the min frequency changes again.  Any idea how I can disable that automatic change?  PS: If you need more details, just tell me where to find them and I will provide them Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is real-valued and defined on a bounded interval. How to smooth shade an object while retaining hard edges? AUC and class imbalance in training/test dataset I just start to learn the Area under the ROC curve (AUC). I am told that AUC is not reflected by data imbalance. I think it means that AUC is insensitive to imbalance in test data, rather than imbalance in training data.  In other words, only changing the distribution of positive and negative classes in the test data, the AUC value may not change much. But if we change the distribution in the training data, the AUC value may largely change. The reason is that the classifier cannot be learned well. In this case, we have to use undersampling and oversampling. Am I right? I just want to make sure my understanding on AUC is correct. Where is the right place to register/enqueue scripts & styles Using headset controls like a smart phone What's the difference between iterating over a file with foreach or while in Perl? Suppose that $f$ is holomorphic on $\mathbb C$ and that $\Re(f(z))\ge 0$ for all $z$. Show that $f$ is constant. [Hint: consider $e^{−f(z)}$.]   My thoughts: If $\Re(f(z))\ge 0 $ holds, then $e^{−f(z)}$ is a bounded holomorphic function (do I need to prove this or is it obvious?) So then by Liouville's Theorem $e^{−f(z)}$ is constant.   But then I'm not sure how to rigorously go from this $\exp(−f(z))$ back to $f(z)$.   Could anyone help me piece this together please?  Thanks How do I run executable scripts in Nautilus? What is the relationship between ISO, aperture, and shutter speed? I know digital cameras have ISO options, and that ISO is the camera's sensitivity to light, but if you set  higher ISO then you can get a noisy image. I also know there are two other camera options, shutter speed and aperture.   What is the relationship of between them? Is there any equation, or something like that?  For example, if I set the ISO to 640, then how should I set the shutter speed and aperture? 80's Animated movie: An old man joins mission to save a boy stranded on a hostile world What is the reason behind the multiple Spider-Man reboots? Spider-Man is the one franchise that is on course to a third reboot in a span of 15 years.   My question is what was the reason for these numerous reboots of Spider-Man which pretty much follow the same origin story for the Marvel superhero.  Has the comic creator or studio ever spoken out about the reasoning behind the same? Transit visa for Indian citizen traveling through Hong Kong What's the fastest and easiest to read implementation of calculating the sum of digits?  I.e. Given the number: 17463 = 1 + 7 + 4 + 6 + 3 = 21 How to change the list of file types which Safari will open immediately after download? I would like to change Safari Safe List of type files which open immediately after download. Any ideas how? Novel about a teleportation device and a con artist alien