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!

Plotting – Riemann surface incomplete gamma function

Michael Trott has shown in his book The Mathematica GuideBook for Symbolics, page 1003, a good way to visualize the Riemann surface of the incomplete gamma function $ Gamma ( alpha, z) $, To illustrate the Riemann surface of example $ Gamma ((3 + i) / 10, z) $, we have:

With({(Alpha) = 0.3 + 0.1 I, ee = 10^(-12)}, 
Show(GraphicsArray(
Graphics3D({EdgeForm(Thickness(0.002)), 
SurfaceColor(Hue(0.09), Hue(0.18), 2.3), 
Table((*split whole graphics into pieces*)
Last /@ Partition(
Cases(ParametricPlot3D((*the sheets*){r Cos((CurlyPhi)), 
r Sin((CurlyPhi)), #(
Exp(2 k Pi I (Alpha)) Gamma((Alpha), 
r Exp(I (CurlyPhi))) + (1 - 
Exp(2 k Pi I (Alpha))) Gamma((Alpha)))}, {r, 0, 
2}, {(CurlyPhi), -Pi + ee , Pi - ee}, 
PlotPoints -> {30, 40}, 
DisplayFunction -> Identity), _Polygon, Infinity), 
2), {k, -2, 2})}, BoxRatios -> {1, 1, 2.5}, 
PlotRange -> All) & /@(*show real and imaginary part*){Re, 
Im})))

where we used the identity: $ Gamma ( alpha, exp (2 k pii) z) = exp (2 k pii alpha) Gamma ( alpha, z) + (1-exp (2 k pii alpha) Gamma ( alpha)) $,

Mathematica does not return anything. Every help is appreciated!

Probability – Which random walk can produce a gamma distribution in the border area?

Symmetrical random walk, its probability distribution is the binomial coefficient, in the continuous boundary the Gaussian distribution:

$ displaystyle e ^ {- x ^ {2}} $

What kind of random walk is the probability distribution in the limit of the gamma distribution:

$ displaystyle xe ^ {- x} $ to the $ x geqslant 0 $ ?

or simpler, an exponential distribution:

$ displaystyle e ^ {- x} $ to the $ x geqslant 0 $ ?

We are looking for something as simple as a random walk at a certain time, in the steady limit, exponential factor $ e ^ {- x} $ Factor shows up. At each step, the rules for casual walking should be as simple as possible. If possible, we want each step of the random walk to be an iid random variable (independent and identically distributed). Is that possible ?

Thanks.

at.algebraic topology – Is the inclusion $ Delta ^ {op} to Gamma ^ {op} = Fin_ ast $ homotopy cofinal?

There is a canonical functor $ i: Delta ^ {op} to Fin_ ast $, For example, one uses the pullback $ i ^ ast $ turn on $ Gamma $space into a simplicial space and then make a geometric realization to get a delooping.

Question: Is the functor $ i $ Homotopy cofinal?

That is, can I calculate the homotopy colimit of a? $ Gamma $-space, viewed as a functor $ Fin_ ast to Top $by precomposing with $ i $ and take the geometric realization of the emerging simplicial space? Equivalent (from the $ infty $-Categorical Quill Set A) are the cosice categories $ langle n rangle downarrow i $ weakly contractible for everyone $ langle n rangle in Fin_ ast $?

I especially like a description of $ i $ I learned that from a work by Ayala, Francis and Tanaka. think of $ Delta $ as an incomplete subcategory of the category of one-dimensionally layered spaces and layered maps. Then $ i $ is the functor that sends a CW complex $ (n) $ to his sentence $ langle n rangle $ of one-dimensional layers plus a disjoint base point. A morphism $ f $ will be sent to the card $ f ^ ast $ Transferring a one-dimensional layer to the one-dimensional layer (if any) under which it is imaged $ f $and otherwise the base point.

Sequences and Series – Is there a deep philosophy or intuition behind the similarity between $ pi / 4 $ and $ e ^ {- gamma} $?

Here are a few examples of the similarity of Wikipedia, where the expressions differ only in characters.
I also came across other analogies.

$$ { begin {align} gamma & = int _ {0} ^ {1} int _ {0} ^ {1} { frac {x-1} {(1-xy) ln xy} } , dx , dy \ & = sum_ {n = 1} ^ { infty} left ({ frac {1} {n}} – ln { frac {n + 1} {n }} right). end {align}} $$

$$ { begin {align} ln { frac {4} { pi}} & = int _ {0} ^ {1} int _ {0} ^ {1} { frac {x-1 } {(1 + xy) ln xy}} , dx , dy \ & = sum_ {n = 1} ^ { infty} left ((- 1) ^ {n-1} left ({ frac {1} {n}} – ln { frac {n + 1} {n}} right) right). end {align}} $$

$$ { begin {align} gamma & = sum_ {n = 1} ^ { infty} { frac {N_ {1} (n) + N_ {0} (n)} {2n (2n + 1)}} \ ln { frac {4} { pi}} & = sum_ {n = 1} ^ { infty} { frac {N_ {1} (n) -N_ {0} (n)} {2n (2n + 1)}}, end {align}} $$

I wonder if there is an algebraic system there $ 4e ^ {- gamma} $ would play a similar role as what $ pi $ plays, for example in complex numbers, or in a geometric system in which $ 4e ^ {- gamma} $ would play a special role, like $ pi $ in Euclidean and Riemannian geometries.

Solving a polynomial equation with gamma function

I would recommend using the new M12 feature AsymptoticSolve for this. Your equation:

eqn = FD1((d-2)/2, ηs) + FD1((d-2)/2, ηs - vd) == 2 FD1((d-2)/2, η0);

We have to find the zero-order approximation of ηs when vd is small:

Simplify(Solve(eqn /. vd -> 0), (η0 | vd) ∈ Reals)

{{ηs -> η0}}

Use now AsymptoticSolve:

AsymptoticSolve(eqn, {ηs, η0}, {vd, 0, 5})

{{ηs ->
vd / 2 + ((-6 + d) (-2 + d) (-1 + d) vd ^ 4) / (
1536 η0 ^ 3) + ((2-d) vd ^ 2) / (16 η0) + η0}}

supplement

If you have an earlier version of Mathematica, you do not have access to it AsymptoticSolveYou could try using the cloud instead. Define for example:

asymptoticSolve(args__) := CloudEvaluate(System`AsymptoticSolve(args))

Then use asymptoticSolve instead of AsymptoticSolve,

Algorithms – Combinatorial optimization, how to choose the optimal gamma distribution?

Configuration: To let $ A = {X_1, …, X_n } $ independent, but not necessarily identical, Bernoulli random variables. Let's say you get a bunch of $ m + 1 $ weights $ W = {w_0, w_1, …, w_m } $With $ m leq n $, and $ -1 leq w_k leq 1 $ for all $ 0 leq k leq m $, The goal is to select a set $ S subset A $ from $ m $ Random variables, say $ S = {X_ {s_1}, …, X_ {s_m} } $that maximizes

$$ sum_ {i = 0} ^ m ~ w_i mathbb {P} bigg ( sum_ {j = 1} ^ m X_ {s_j} = i bigg) $$

Question: Can the optimal solution or even a constant factor approximation in polynomial time be calculated?

Previous work: If we have both $ w_0 leq w_1 leq … leq w_m $ or $ w_0 geq w_1 geq … geq w_m $Then we can achieve the optimal solution by either the $ X_i $& S; s with the highest probability of giving 1 or the lowest probability.
But does it work for a general? $ W $ limited by $ -1 $ and $ 1 $?

Graphics Programming – Additive Blending and Gamma Correction

Should additive blending (also known as light mapping) be done in linear space?

I tried to do it in linear space, and it got linear and boring and lost the cool HDR-like Bloomy effect. Is there a standard method for additive mixing in linear RGB, or is additive mixing now an outdated gamma-space hack and should be forgotten?