ct.category theory – Functor category with only isomorphisms in the Gamma space construction

I have not read Marcolli’s paper, but it sounds like the construction you are describing goes back to Segal in “Categories and cohomology theories”.

I think the short answer to your question is people take the category of isomorphisms because in this way one gets the most interesting homotopy type. Suppose for example you took the category of summing functors and all natural transformations between them. Since the category $C$ has has a zero object, it follows that the category of summing functors has an initial object, so the geometric realization of the category of summing functors would be contractible.

Segal’s motivation was to construct interesting infinite loop spaces out of categories. The idea (or rather my crude simplification of it!) is that if $C$ is a symmetric monoidal category, then its classifying space $BC$ is not quite an infinite loop space, but it is close: it is an infinite loop space after group completion. The category of summable functors from a set with $n$-elements to $C$ is a model for the $n$-fold power $C^n$, that is suitable for constructing a delooping (of the group completion) of $BC$.

One way to get a symmetric monoidal category is to start with a category with coproducts and then take its full subcategory of isomorphisms. For some reason it turns out that some of the most interesting infinite loop spaces are obtained in this way: the first space of the sphere spectrum colim$_n Omega^nS^n$ is obtained from the category of finite sets and bijections. The $K$-theory space $mathbb Z times BO$ is obtained from the category of vector spaces and isomorphisms, etc. However, there also are interesting examples of infinite loop spaces that are obtained as the group completion of a symmetric monoidal category where not all morphisms are isomorphisms. Various categories of cobordisms come to mind.

reference request – Book recommendation: The Gamma function

I don’t know if Math Stackexchange is for such questions (probably, I can ask such questions since there is a tag on it) but I want a book on the Gamma function, that is similar to, say, Titchmarsh’s Theory of the Riemann zeta function. I want a book that studies the gamma function in depth. So please recommend a book on the Gamma function that not just lists some properties of the gamma function. I can search Google for it, but I am not sure.

gamma function – algebraic manipulation on equation

An equation involving the poisson point process is formulated as:

$prod_{j=1}^{K}exp(-2pilambda_j(sp_j)^{2/alpha}int_0^{infty}rint_0^{infty}e^{-t(1+r^{alpha})}dtdr)$.

Some algebraic manipulations are carried out and the equation is rewritten as:

$exp(-s^{2/alpha}C(alpha)sum_{i=1}^{K}lambda_iP_i^{2/alpha})$, where $C(alpha)=frac{2pi csc(frac{2}{alpha})}{alpha}$.

I only know the manipulation is related to Gamma function. I want to know the datails about the manipulation.

Thanks a lot! : )

pr.probability – Inaccurate results for the analytical expression of $mathbb{E}left[ a mathcal{Q} left( sqrt{b } gamma right) right]$

I’m trying to plot a graph for the following expectation

$$mathbb{E}left( a mathcal{Q} left( sqrt{b } gamma right) right)=a 2^{-frac{kappa }{2}-1} b^{-frac{kappa }{2}} theta ^{-kappa } left(frac{, _2F_2left(frac{kappa }{2}+frac{1}{2},frac{kappa }{2};frac{1}{2},frac{kappa }{2}+1;frac{1}{2 b theta ^2}right)}{Gamma left(frac{kappa }{2}+1right)}-frac{kappa , _2F_2left(frac{kappa }{2}+frac{1}{2},frac{kappa }{2}+1;frac{3}{2},frac{kappa }{2}+frac{3}{2};frac{1}{2 b theta ^2}right)}{sqrt{2} sqrt{b} theta Gamma left(frac{kappa +3}{2}right)}right)$$
where $a$ and $b$ are constant values, $mathcal{Q}$ is the Gaussian Q-function, which is defined as $mathcal{Q}(x) = frac{1}{sqrt{2 pi}}int_{x}^{infty} e^{-u^2/2}du$ and $gamma$ is a random variable with Gamma distribition, i.e., $f_{gamma}(y) sim frac{1}{Gamma(kappa)theta^{kappa}} y^{kappa-1} e^{-y/theta} $ with $kappa > 0$ and $theta > 0$.

This equation was also found with Mathematica, so it seems to be correct.

Follows some examples, where I have checked the analytical results against the simulated ones.

When $kappa = 12.85$, $theta = 0.533397$, $a=3$ and $b = 1/5$ it returns the correct value $0.0218116$.

When $kappa = 12.85$, $theta = 0.475391$, $a=3$ and $b = 1/5$ it returns the correct value $0.0408816$.

When $kappa = 12.85$, $theta = 0.423692$, $a=3$ and $b = 1/5$ it returns the value $-1.49831$, which is negative. However, the correct result should be a value around $0.0585$.

When $kappa = 12.85$, $theta = 0.336551$, $a=3$ and $b = 1/5$ it returns the value $630902$. However, the correct result should be a value around $0.1277$.

Therefore, the issue happens as $theta$ decreases. For values of $theta > 0.423692$ the analytical matches the simulated results. The issue only happens when $theta <= 0.423692$.

I’d like to know if that is an accuracy issue or if I’m missing something here and if there is a way to correctly plot a graph that matches the simulation.

calculus and analysis – Integral of $r frac{2^{r-1} log (2) e^{-frac{sqrt{2^r-1}}{b}} left(2^r-1right)^{frac{d}{2}-1}}{b^d Gamma (d)}$ with Mathematica

I’m trying to find the integral given below with Mathematica

$int_0^{infty } r frac{2^{r-1} log (2) e^{-frac{sqrt{2^r-1}}{b}} left(2^r-1right)^{frac{d}{2}-1}}{b^d Gamma (d)} , dr$

However, it takes too long for it to return something and when it returns it outputs the same integral.

$int_{0}^{infty } frac{2^{r-1} r log (2) b^{-d} e^{-frac{sqrt{2^r-1}}{b}} left(2^r-1right)^{frac{d}{2}-1}}{Gamma (d)} , dr$

I’d like to figure out the solution for this integral.

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:

data 1

A picture of how data 2 looks is here:

data 2

Thank you in advanced,