Mathematica does not recognize this case:
Integrate((T^a/(Gamma(a))) * (E^(-Tx)) (x^(a - 1)), {x, 0, Infinity})
Integrate::idiv: Integral of (E^-Tx T^a x^(-1+a))/Gamma(a) does not converge on {0,(Infinity)}.
This should converge because
Tag: gamma
coding theory – Variance Gamma parameter estimation in R studio
I am using the vgFit function in R Studio to estimate parameters for stock prices. I converted the stock prices to a vector and I get this error. My code was:
param<-c(0,1,0,1)
fit<-vgFit(c(x), param=param)
The error is:
Error in optim(paramStart, llsklp, NULL, method = startMethodSL, hessian = FALSE, :
function cannot be evaluated at initial parameters
analytic number theory – Is every modular function on $Gamma$ univalent?
Suppose $f$ is a modular function on $Gamma$ then it has a fundamental region $R_L$. Since $f$ is modular, it can be expressed as a rational function of $J(tau)=frac{g_2^3(tau)}{Delta(tau)}$. To prove this we actually construct a rational function
$$g(tau)=frac{prod(J(a_i)-J(tau))}{prod(J(b_i)-J(tau))}$$
where the $a_i,b_i$ are zeros and poles of $f$. It seems to me that this construction implies the possibility of having multiple poles in the fundamental region. I know for sure that $J$ has only one pole since it attains every value exactly once.
Since a modular function is univalent if and only if its fundamental region has genus $1$ and if we identify the edges of $R_Gamma$ it would become homeomorphic to $mathbb{P}^1$, any modular function that has the fundamental region $R_Gamma$ should be univalent, that is, any modular function is univalent in the closure of its fundamental region.
This really confused me and I am not sure if this is correct.
Fitting data to left skewed gamma distribution
How can I fit the following two set of data to a left skewed gamma function, which I what I think should fit the data best?:
data 1 is here: https://pastebin.com/X2HTjTP7
data 2 is here: https://pastebin.com/8Rh4BHDT
Is there any other suggestion of what would be the best distribution or equation to fit the data?
A picture of how data 1 looks is here:
A picture of how data 2 looks is here:
Thank you in advanced,
simplifying expressions – Is there any way to prioritize Gamma function?
In Mathematica, if you type
Gamma[1, 0, -a]
It automatically simplifies to 1 - E^a. I wonder though if there is a way to stop Mathematica from simplify Gamma function automatically.
I understand that I can use Inactive or HoldFrom. But I actually want having to remember using Inactive every time.
How to calculate gamma function values for quaternions with Mathematica?
How can we calculate the values for that? gamma Function with Quaternions on Mathematica?
For example:
Gamma(5) = 24
N(Gamma(I)) = -0.15495 - 0.498016 I
N(Gamma(1 + 2 I)) = 0.151904 + 0.0198049 I
N(Gamma(Quaternion(5, 0, 0, 0))) = Gamma(Quaternion(5., 0., 0., 0.)) (* which is wrong *)
How can we calculate specifically?
Gamma(Quaternion(1, 2, 3, 4))?
Tries:
N(Gamma(FromQuaternion(Quaternion(1, 2, 0, 0)))) (* works *)
N(Gamma(FromQuaternion(Quaternion(1, 2, 3, 4)))) (* doesn't work *)
QuaternionPower(x_, y_) := E^(y ** Log(x))
QuaternionGamma(z_) := Integrate(QuaternionPower(E, -x + Log(x)*(z - 1)), {x, 0, Infinity})
N(QuaternionGamma(FromQuaternion(Quaternion(1, 2, 0, 0)))) = 0.151904 + 0.0198049 I (* works *)
N(QuaternionGamma(FromQuaternion(Quaternion(1, 2, 3, 0)))) = ConditionalExpression(Gamma((1. + 2. I) + 3. J), Re(J) > -0.333333) (* doesn't work *)
Logic – If a set of sentences $ Gamma $ has an infinite model, does it also have a denumerable model?
I think that's wrong, since we can only conclude from the Downward-Lowenheim-Skolem theorem $ Gamma $ has an enumeration model, it is not shown whether this enumeration model is infinite. Certainly ours $ Gamma $ already has an infinite model, but nothing says that its enumerable model must be identical to its infinite model.
I can't see how this claim is true, but I think it should be true?
Gamma coin round 3 air drops | Proxies-free
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Optimization – Maximize $ log (4) c + log (3) a + log (2) x $ if $ a + c + x + y = 1 $, $ (a + c) ^ 2 + (x + y) ^ 2 + 2xc leq 1-2 gamma $, $ 0 leq gamma leq 1/4 $
I want to find out how to show by hand that the maximum of
$$ log (4) c + log (3) a + log (2) x $$
when
$$ a geq 0, c geq 0, x geq 0, y geq 0, $$
$$ a + c + x + y = 1, $$
$$ (a + c) ^ 2 + (x + y) ^ 2 + 2xc leq 1-2 gamma, $$
Where $ gamma $ is a fixed constant so that $ 0 leq gamma leq 1/4 $.
Maple showed me that the maximum value is reached at $ frac { log (6)} {2} + frac { sqrt {1-4 gamma}} {2} log (3/2) $that is given by $ a = frac {1+ sqrt {1-4 gamma}} {2} $, $ b = frac {1- sqrt {1-4 gamma}} {2} $, $ c = y = 0 $.
I've tried using Lagrangian multipliers by changing the last constraint to an equality, but it gets very messy and I'm stuck.
I would also be happy if I could just show it $ log (4) c + log (3) a + log (2) x leq frac { log (6)} {2} + frac { sqrt {1-4 gamma}} { 2} log (3/2) $ for all such $ a, c, x, y $. One method I've tried is to use the concavity of $ log (x) $. The equation of the line that goes through $ (2, log (2)) $ and $ (3, log (3)) $ is $ L (x) = log (3/2) x- log (9/8) $. Then by concavity of $ log (x) $ we have $ log (x) leq L (x) $ for all integers $ x $. Then
begin {align *} & log (4) c + log (3) a + log (2) x \
& leq L (4) c + L (3) a + L (2) x \
& = log (3/2) (4c + 3a + 2x + y) – log (9/8) \
& = log (3/2) left ( frac {5 + 3} {2} c + frac {5 + 1} {2} a + frac {5-1} {2} x + frac {5- 3} {2} y right) – log (9/8) \
& = log (3/2) left ( frac {5+ sqrt {1-4 gamma}} {2} right) – log (9/8) + log (3/2) left ( frac {ax + 3 (cy) – sqrt {1-4 gamma}} {2} right) \
& = frac { log (6)} {2} + frac { sqrt {1-4 gamma}} {2} + log (3/2) left ( frac {ax + 3) ( cy) – sqrt {1-4 gamma}} {2} right). \
end {align *}
That is, to get the inequality I want, it would be enough to show it $ a-x leq 3 (y-c) + sqrt {1-4 gamma} $. However, I have not been able to prove this inequality. Any suggestions or help will be greatly appreciated.
Another option would be to use jamming techniques, but I have no experience with them. Thank you for your help.
Statistics – probability between two values of a gamma distribution
I'm trying to solve a problem where I have a random variable X that follows a gamma distribution of $ Gamma ( theta = 5, alpha = 4) $.
I'm trying to find the probability of $ P (10 leq X leq 30) $. Of course, if this were a normal distribution, it would be easy, but I currently don't know how to deal with this problem. I am confused as to whether to convert it to one $ chi ^ 2 $ and how that helps me to solve this problem. Thank you very much