Command line – xrandr could not determine the size of the gamma for standard output

I use Ubuntu 19.04 (code name: Disco) Intel G2030 4 GB DDR3 Ram no GPU (only Intel onboard graphics) that I have not installed any The drivers just ticked "Install third party software for graphics and Wi-Fi hardware and additional media formats" when installing Ubuntu (and when booting between Ubuntu and Windows 10) when trying to set a custom resolution for my display

:~$ xrandr
xrandr: Failed to get size of gamma for output default
Screen 0: minimum 640 x 480, current 1366 x 768, maximum 1368 x 768
default connected primary 1366x768+0+0 0mm x 0mm
   1366x768       0.00* 
   640x480        0.00  
   1368x768       0.00  
:~$ cvt 1600 900 60
# 1600x900 59.95 Hz (CVT 1.44M9) hsync: 55.99 kHz; pclk: 118.25 MHz
Modeline "1600x900_60.00"  118.25  1600 1696 1856 2112  900 903 908 934 -hsync +vsync
:~$ xrandr --newmode "1600x900_60.00"  118.25  1600 1696 1856 2112  900 903 908 934 -hsync +vsync
xrandr: Failed to get size of gamma for output default
:~$ xrandr --addmode default "1600x900_60.00"
xrandr: Failed to get size of gamma for output default

Then the option becomes available in the resolution menu, but if I change it, nothing happens, it stays the same

analytical number theory – inequalities $ pi (x ^ a + y ^ b) ^ alpha leq pi (x ^ c) ^ beta + pi (y ^ d) ^ gamma $ including the prime function, where the Constants are very close to $ 1 $

To let $ pi (x) $ Be the prime count function, I'm curious whether a suitable variant of the second Hardy-Littlewood conjecture (this corresponding Wikipedia)
$$ pi (x ^ a + y ^ b) ^ alpha leq pi (x ^ c) ^ beta + pi (y ^ d) ^ gamma tag {1} $$
can be proven where the constants $ 0 <a, b, c, d leq 1 $ and the constants $ 0 < alpha, beta, gamma leq 1 $ are very close to our ceiling $ 1 $for all real numbers $ x <y $ With $ L <x $ for a suitable choice of a constant $ L $,

Question. Is it possible to prove any statement of the type $ (1) $ under the mentioned conditions for constants $ 0 <a, b, c, d leq 1 $ and constants $ 0 < alpha, beta, gamma leq 1 $ all of these (all together /
at the same time) very close to $ 1 $for all real numbers $ x <y $ for a suitable one $ L <x $? Many thanks.

I don't know if that kind of suggestions $ (1) $ are in the literature or are essentially the same original second Hardy-Littlewood conjecture when we request that these constants be very close $ 1 $,

If there is relevant literature, answer my question as a reference request and I will try to find and read these statements from the literature.

references:

(1) G.H. Hardy and J.E. Littlewood, Some problems of “Partitio numerorum” III: About the expression of a number as a sum of prime numbersActa Math. (44): 1-70 (1923).

Abstract algebra – General method for determining $ mathbb {Q} ( gamma) = mathbb {Q} ( alpha, beta) $ by specifying $ alpha $ and $ beta $

I am currently reading S. Langs "Undergraduate algebra". According to the primitive root element theorem (field theory chapter), there are a number of exercises to find a primitive element of extensions and then their degrees. However, I don't even know how to start. They are as below:

  1. Find one item at a time $ gamma $ so that $ mathbb {Q} ( alpha, beta) = mathbb {Q} ( gamma) $, Prove every statement you make.

on) $ alpha = sqrt {-5}, beta = sqrt {2} $

b) $ alpha = sqrt (3) {2}, beta = sqrt {2} $

c) $ alpha = $ Root of $ t ^ 3 -t + 1 $ . $ beta = $ Root of $ t ^ 2-t-1 $

d) $ alpha = $ Root of $ t ^ 3 -2t + 3 $. $ beta = $ Root of $ t ^ 2 + t + 2 $

$ quad $2. Find the degrees of the fields $ mathbb {Q} ( alpha, beta) $ over $ mathbb {Q} $ in any case from exercise 1.

I think exercises a) and b) are pretty much the same, but I'm not sure about c) and d).

Repeat identity of the gamma function is evaluated as "False"

Sorry, if this is a duplicate. There are many questions related to gamma functions, and I have not done a comprehensive search.

I go through a textbook in which the gamma function is developed, and they begin by stating a repeat relationship that every faculty extension must meet:

$$
f (x) = x f (x-1); f (0) = 1
$$

They then introduce the definition of the gamma function and show that it satisfies the relationship after a slight shift of the coordinate frame.

$$
Gamma (x + 1) = x gamma (x)
$$

I like to check how Mathematica simplifies expressions to detect any quirks. I was surprised when I received the following result:

Assuming(x > 0 && x (Element) Reals,TrueQ(x Gamma(x) == Gamma(x + 1)))
(*False*)

I've checked the documentation to see if the definitions between the book and MMA do not match, but that does not seem to be the case here.

My question: is the expected behavior? If so, can you please tell me what I miss?

I did this on a clean kernel and get the same result.

$Version
(*12.0.0 for Linux x86 (64-bit) (April 7, 2019)*)

Appreciate all help.

symbolic – Closed form of the product of the gamma function

Mathematica
recognizes this closed form
begin {align}
prod_ {k = 1} ^ {n-1} sin ( pi k / n)
& =
2 ^ {1-n} , n
end

All good:

Enter image description here

but fails
on this

Enter image description here

Nevertheless, this expression also has a known closed form
begin {align}
prod_ {k = 1} ^ {n-1} gamma (k / n)
& =
sqrt { frac {(2 , pi) ^ {n-1}} {n}}
,
end

Is there a way to do Mathematica
to recognize it?

Differential Geometry – Is there a simple closed-plane curve with $ pi ( gamma? (T)) ge 0 $ for the projection $ pi (x, y) = x $?

To let $ pi: mathbb {R} ^ {2} to mathbb {R} $ be the projection of $ pi (x, y) = x $,
Is there a simple curve with a closed plane? $ gamma: I to mathbb {R} ^ {2} $ so for all $ t in I $, we have $ pi ( gamma & # 39; t) ge 0 $?

I tried to define the reparmization of $ gamma $ such as $ tilde { gamma} $ and tried to use Hopf's circulation set, but could not do it. Can you help?

Plotting – How can the evaluation of a sum of incomplete gamma functions be accelerated?

For the following function we need to create a ContourPlot $ H (x, y, n) $:

f(x_,y_,k_,n_,c_) := 
Gamma(c + (I/2)(x + I*y), k^2*(Pi/n), 
   k^2*Pi n)/(k^2*Pi)^(1/4 + (I/2)*(x + I*y));

g(x_,y_,k_,n_,c_) := f(x,y,k,n,c) + f(-x,-y,k,n,c);

H(x_,y_,n_) := 
 Sum((3/2)*g(x,y,k,n,5/4) - g(x,y,k,n,9/4), {k, 1, n});

The command for the contour representation is:

n = 6;
ContourPlot({Re(H(x, y, n)) == 0, 
Im(H(x, y, n)) == 0}, {x,-1,35}, {y,-1,35}, 
AxesLabel -> {"x", "y"}, PlotPoints -> 50)

The problem we have is for bigger ones $ n $, to like $ n = 12 $It takes too long (more than 30 minutes on my laptop (Intel i7 CPU, 16G memory) with Mathematica 11.3) to finish the plot.

question

Is there a way in Mathematica to speed up the evaluation and drawing of such functions?

Many thanks!