The group of automorphisms of the first weyl algebra in prime characteristic and the restriction map

Let K be a perfect field of characteristic p > 0; A(1) := K be the first Weyl algebra; and Z := K[X := x(p), Y := partial derivative(p)] be its centre. It is proved that (1) the restriction map res : Aut(K)(A(1)) -> Aut(K)(Z), sigma bar right arrow sigma vertical bar(Z) is a monomorphism with im(res) = Gamma := (tau is an element of Aut(K)(Z) vertical bar J(tau) = 1), where J(tau) is the Jacobian of tau, (Note that Aut(K)(Z) = K* (sic) Gamma, and if K is not perfect then im(res) not equal Gamma.); (ii) the bijection res : Aut(K)(A(1)) -> Gamma is a monomorphism of infinite dimensional algebraic groups which is not an isomorphism (even if K is algebraically closed); (iii) an explicit formula for res(-1) is found via differential operators D(Z) on Z and negative powers of the Fronenius map F. Proofs are based on the following (non-obvious) equality proved in the paper: (d/dx + f)(p) = (d/dx)(p) + d(p-1)f/dx(p-1) + f(p), f is an element of K[x]

Remarks on a normal subgroup of GA_n

We show that the subgroup generated by locally finite polynomial automorphisms of k^n is normal in GA_n. Also, some properties of normal subgroups of GA_n containing all diagonal automorphisms are given.Comment: 5 page

Hamiltonian Formalism in Quantum Mechanics

Heisenberg motion equations in Quantum mechanics can be put into the Hamilton form. The difference between the commutator and its principal part, the Poisson bracket, can be accounted for exactly. Canonical transformations in Quantum mechanics are not, or at least not what they appear to be; their properties are formulated in a series of Conjectures

On the lifting of the Nagata automorphism

It is proved that the Nagata automorphism (Nagata coordinates, respectively) of the polynomial algebra $F[x,y,z]$ over a field $F$ cannot be lifted to a $z$-automorphism ($z$-coordinate, respectively) of the free associative algebra $K$. The proof is based on the following two new results which have their own interests: degree estimate of ${Q*_FF}$ and tameness of the automorphism group ${\text{Aut}_Q(Q*_FF)}$.Comment: 15 page

Implementation of Trust Issues in Ecommerce

This paper is based on security of E-commerce web sites. It uses web mining technology for providing security on e-commerce web sites. The connection between web mining security and ecommerce analyzed based on user behavior on web . Different web mining algorithms and security algorithm are used to provide security on e-commerce web sites. Based on customer behavior web mining algorithms like page rank algorithm and trust rank algorithm is used for developing web mining framework in e-commerce web sites. This application will develop false hit database algorithm and nearest neighbor algorithm to provide security on e-commerce web site. It will also develop page rank & trust rank algorithm for providing security on e-commerce site

Affine modifications and affine hypersurfaces with a very transitive automorphism group

We study a kind of modification of an affine domain which produces another affine domain. First appeared in passing in the basic paper of O. Zariski (1942), it was further considered by E.D. Davis (1967). The first named author applied its geometric counterpart to construct contractible smooth affine varieties non-isomorphic to Euclidean spaces. Here we provide certain conditions which guarantee preservation of the topology under a modification. As an application, we show that the group of biregular automorphisms of the affine hypersurface $X \subset C^{k+2}$ given by the equation $uv=p(x_1,...,x_k)$ where $p \in C[x_1,...,x_k],$ acts $m-$transitively on the smooth part reg$X$ of $X$ for any $m \in N.$ We present examples of such hypersurfaces diffeomorphic to Euclidean spaces.Comment: 39 Pages, LaTeX; a revised version with minor changes and correction

IT Risks in Measure and Number

Verhoef, C. [Promotor]Peters, R.J. [Copromotor

Metabolic evolution and the self-organization of ecosystems

Metabolism mediates the flow of matter and energy through the biosphere. We examined how metabolic evolution shapes ecosystems by reconstructing it in the globally abundant oceanic phytoplankter Prochlorococcus To understand what drove observed evolutionary patterns, we interpreted them in the context of its population dynamics, growth rate, and light adaptation, and the size and macromolecular and elemental composition of cells. This multilevel view suggests that, over the course of evolution, there was a steady increase in Prochlorococcus' metabolic rate and excretion of organic carbon. We derived a mathematical framework that suggests these adaptations lower the minimal subsistence nutrient concentration of cells, which results in a drawdown of nutrients in oceanic surface waters. This, in turn, increases total ecosystem biomass and promotes the coevolution of all cells in the ecosystem. Additional reconstructions suggest that Prochlorococcus and the dominant cooccurring heterotrophic bacterium SAR11 form a coevolved mutualism that maximizes their collective metabolic rate by recycling organic carbon through complementary excretion and uptake pathways. Moreover, the metabolic codependencies of Prochlorococcus and SAR11 are highly similar to those of chloroplasts and mitochondria within plant cells. These observations lead us to propose a general theory relating metabolic evolution to the self-amplification and self-organization of the biosphere. We discuss the implications of this framework for the evolution of Earth's biogeochemical cycles and the rise of atmospheric oxygen.Simons Foundation (Grant SCOPE 329108)Gordon and Betty Moore Foundation (Grant 3778)Gordon and Betty Moore Foundation (Grant 495.01