Let A denote the ring of differential operators on the affine line with its two usual generators t and d/dt given degrees +1 and -1 respectively. Let X be the stack having coarse moduli space the affine line Spec k[z] and isotropy groups Z/2 at each
integer point. Then the category of graded A-modules is equivalent to the category of quasi-coherent sheaves on X. Version 2: corrected typos and deleted appendix at referees suggestion.
We compute the Grothendieck and Picard groups of a complete smooth toric Deligne-Mumford stack by using a suitable category of graded modules over a polynomial ring.
In this paper we tried a different approach to work out the integrals of e^(x^n) and e^(-x^n). Integration by parts shows a nice pattern which can be reduced to a form of series. We have shown both the indefinite and definite integrals of the functio
ns mentioned along with some essential properties e.g. conditions of convergence of the series. Further more, we used the integrals in form of series to find out series solution of differential equations of the form x[(d^2 y)/(dx^2)]-(n-1)(dy/dx)-n^2 x^(2n-1)y-nx^n=0 and x[(d^2 y)/(dx^2)] -(n-1)(dy/dx)-n^2x^(2n-1)y+(n-1)=0, using some non standard method. We introduced modified Normal distribution incorporating some properties derived from the above integrals and defined a generalized version of Skewness and Kurtosis. Finally we extended Starlings approximation to limit [n to infinity ] (2n)! ~ 2n * sqrt{(2pi)} [(2n/e)]^(2n).
Wattss Theorem says that a right exact functor F:Mod R-->Mod S that commutes with direct sums is isomorphic to -otimes_R B where B is the R-S-bimodule FR. The main result in this paper is the following: if A is a cocomplete abelian category and F:Mod
R --> A is a right exact functor commuting with direct sums, then F is isomorphic to - otimes_R B where B is a suitable R-module in A, i.e., a pair (B,f) consisting of an object B in A and a ring homomorphism f:R --> Hom_A(B,B). Part of the point is to give meaning to the notation -otimes_R B. That is done in the paper by Artin and Zhang on Abstract Hilbert Schemes. The present paper is a natural extension of some of the ideas in the first part of their paper.
