Abstract We introduce the notion of approximation type for partial, and in certain cases total description extensions a given valuation from field K to rational function ( x ). To every extension, unique over is associated, while may be limit many pseudo Cauchy sequences. Approximation types also provide information where are not immediate, we prove that they correspond bijectively when algebra...

Let P∈Q[t,x] be a polynomial in two variables with rational coefficients, and let G the Galois group of P over field Q(t). It follows from Hilbert's Irreducibility Theorem that for most numbers c specialized P(c,x) has isomorphic to factors same way as P. In this paper we discuss methods computing obtaining an explicit description exceptional c, i.e., those which different or differently To ill...

Gopalakrishnan et al [1] described a method called \growth transform" to optimize rational functions over a domain, which has been found useful to train discriminatively Hidden Markov Models(HMM) in speech recognition [5, 6, 9]. A sum of rational functions is encountered when the contributions from other HMM states are weighted in estimating Gaussian parameters of a state, and the weights are o...

Let F/K be an algebraic function field in one variable over a finite field of constants K, i.e., F is a finite algebraic extension of K(x) where x ∈ F is transcendental over K. Let E be a finite separable extension of F . Let N(E) and g(E) denote the number of places of degree one (or rational places), and the genus, respectively, of E. Let [E : F ] denote the degree of this extension. In recen...

We find low order approximations to the spherical nonreflecting boundary kernel for the wave equation in three dimensions. First we express the Laplace transform of the kernel as a rational function by solving for the zeros of a modified Bessel function. Then we formulate a linear time-invariant dynamical system whose transfer function is this rational function. Finally we use the Balanced Trun...

We give new proofs of two theorems of Stanley on generating functions for the integer points in rational cones. The first, Stanley’s reciprocity theorem, relates the rational generating function σv+K(x) := ∑ m∈v+K∩Z x m, where K is a rational cone and v ∈ R, with σ−v+K◦(1/x). The second theorem asserts that the generating function of the Ehrhart quasipolynomial LP(n) := # (

We give new proofs of three theorems of Stanley on generating functions for the integer points in rational cones. The first, Stanley’s Reciprocity Theorem, relates the rational generating function σv+K(x) := ∑ m∈v+K∩Z x m, where K is a rational cone and v ∈ R, with σ−v+K◦(1/x). The second, Stanley’s Positivity Theorem, asserts that the generating function of the Ehrhart quasipolynomial LP(n) :=...

This work has been contributed on the visualization of curves and surfaces for constrained data. A rational cubic function, with free shape parameters in its description, has been introduced and used. This function has been constrained to visualize the preservation of shape of the data by imposing constraints on free parameters. The rational cubic curve case has also been extended to a rational...