## 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: 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

## Calculate \$ int_ {0} ^ { infty} frac {e ^ {- alpha x} beta ^ x gamma ^ {x -1}} { gamma (x)} dx \$

To let $$alpha, beta, gamma> 0$$ and let $$Gamma$$ denote the gamma function. How can the following integral be calculated?
$$int_ {0} ^ { infty} frac {e ^ {- alpha x} beta ^ x gamma ^ {x -1}} { gamma (x)} dx$$
Is there a useful transformation?

## 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 $$0 and the constants $$0 < alpha, beta, gamma leq 1$$ are very close to our ceiling $$1$$for all real numbers $$x With $$L for a suitable choice of a constant $$L$$,

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

## oa.operator algebras – property gamma for type III factors

I am struggling to find a definition that uses centralizing sequences for property gamma in type III factors.
If $$M$$ is type $$mathrm {III}$$ Factor, what is the exact definition of the property gamma without the equivalent definition: the relative commutant of ultrafilter of $$M$$ in the $$M$$ is not trivial?

## 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:

but fails
on this

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?

## Reference Request – Rejection of Class Group Assignment \$ Gamma_ {g, n} \$

To let $$S_ {g, n}$$ be a Riemann surface of the genus $$g$$With $$n$$ Removed points. The mapping class group of $$S_ {g, n}$$ is designated with $$Gamma_ {g, n}$$,

Is there a reference where the Abelianization of $$Gamma_ {g, n}$$ calculated (or at least for $$g$$ big enough, are they trivial?

## 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!