Generating Sequences and Semigroups of Valuations on 2-Dimensional Normal Local Rings

In this paper we develop a method for constructing generating sequences for valuations dominating the ring of a two dimensional quotient singularity. Suppose that $K$ is an algebraically closed field of characteristic zero, $K[X,Y]$ is a polynomial ring over $K$ and $\nu$ is a rational rank 1 valuation of the field $K(X,Y)$ which dominates $K[X,Y]_{(X,Y)}$. Given a finite Abelian group $H$ acting diagonally on $K[X,Y]$, and a generating sequence of $\nu$ in $K[X,Y]$ whose members are eigenfunctions for the action of $H$, we compute a generating sequence for the invariant ring $K[X,Y]^H$. We use this to compute the semigroup $S^{K[X,Y]^H}$ of values of elements of $K[X,Y]^H$. We further determine when $S^{K[X,Y]} (\nu)$ is a finitely generated $S^{K[X,Y]^H} (\nu)$ -module.Comment: 23 Page

A ruled residue theorem for algebraic function fields of curves of prime degree

The Ruled Residue Theorem asserts that given a ruled extension $(K|k,v)$ of valued fields, the residue field extension is also ruled. In this paper we analyse the failure of this theorem when we set $K$ to be algebraic function fields of certain curves of prime degree $p$, provided $p$ is coprime to the residue characteristic and $k$ contains a primitive $p$-th root of unity. Specifically, we consider function fields of the form $K= k(X)(\sqrt[p]{aX^p+bX+c})$ where $a\neq 0$. We provide necessary conditions for the residue field extension to be non-ruled which are formulated only in terms of the values of the coefficients. This provides a far-reaching generalization of a certain important result regarding non-ruled extensions for function fields of smooth projective conics.Comment: Final version, to appear in Journal of Pure and Applied Algebr

Generating sequences and semigroups of valuations on 2 dimensional normal local rings

In this thesis we develop a method for constructing generating sequences for valuations dominating the ring of a two dimensional quotient singularity. Suppose that K is an algebraically closed field of characteristic zero, K[X, Y] is a polynomial ring over K and v is a rational rank 1 valuation of the field K(X, Y) which dominates K[X, Y](X,Y) . Given a finite Abelian group H acting diagonally on K[X, Y], and a generating sequence of v in K[X, Y] whose members are eigenfunctions for the action of H, we compute a generating sequence for the invariant ring K[X, Y]H. We use this to compute the semigroup SK[X,Y ]H (v) of values of elements of K[X, Y]H. We further determine when SK[X,Y ]H (v) is a finitely generated SK[X,Y ]H (v)-module.Includes bibliographical reference

Modeling optical constants from the absorption of organic thin films using a modified Lorentz oscillator model

Optical constants of organic thin films can be evaluated using the Lorentz oscillator model (LOM) which fails to fit inhomogeneously broadened absorption of highly concentrated molecular films. In modified LOM (MLOM), the inhomogeneous broadening is implemented through a frequency-dependent adjustable broadening function. In this work, we evaluate the optical constants of rhodamine 6G doped poly-vinyl alcohol thin films with varying doping concentration (including also extensively high concentrations) using MLOM, which outperforms LOM by showing a better agreement with the experimental results. Our proposed method provides a way to accurately determine optical constants of isotropic organic thin films only from their absorption spectra without spectroscopic ellipsometry.Peer reviewe

A Design of Digital Microfluidic Biochip along with Structural and Behavioural Features in Triangular Electrode Based Array

AbstractDigital microfluidic based biochip manoeuvres on the theory of microfluidic technology, having a broad variety of applications in chemistry, biology, environmental monitoring, military etc. Being concerned about the technological advancement in this domain, we have focused on equilateral triangular electrodes based DMFB systems. Accepting the associated design issues, here, we have addressed many facets of such electrodes regarding their structural and behavioural issues in comparison to the existing square electrodes. As the requisite voltage reduction is a key challenging design issues, to implement all the tasks using triangular electrodes that are possible in square electrode arrays as well, is a tedious job. Furthermore, to deal with this new design deploying triangular electrodes, we have analyzed all the necessary decisive factors including fluidic constraints to ensure safe droplet movements and other modular operations together with mixing and routing. Moreover, an algorithm has been developed to find a route for a given source and destination pair in this newly designed DMFB. Finally, we have included a comparative study between this new design and the existing one while encountering the above mentioned issues