23 research outputs found
Projective Normality Of Algebraic Curves And Its Application To Surfaces
Let be a very ample line bundle on a smooth curve of genus with
. Then is normally generated if . Let be a triple
covering of genus curve with and a
divisor on with . Then
becomes a very ample line bundle which is normally generated. As an
application, we characterize some smooth projective surfaces.Comment: 7 pages, 1figur
Explicit presentations of nonspecial line bundles and secant spaces
A line bundle L on a smooth curve X is nonspecial if and only if L admits a
presentation L=K_X -D +E for some effective divisors D and E>0 on X with gcd
(D, E)=0 and h^0 (X, O_X (D))=1. In this work, we define a minimal presentation
of L which is minimal with respect to the degree of E among the presentations.
If L=K_X -D +E with degE>2 is a minimal, then L is very ample and any q-points
of X with q <degE are embedded in general position but the points of E are not.
We investigate sufficient conditions on divisors D and E for L=K_X -D +E to be
minimal. Through this, for a number n in some range, it is possible to
construct a nonspecial very ample line bundle L=K_X -D +E on X with/without an
n-secant (n-2)-plane of the embedded curve by taking divisors D and E on X. As
its applications, we construct nonspecial line bundles which show the sharpness
of Green and Lazarsfeld's Conjecture on property (N_p) for general n-gonal
curves and simple multiple coverings of smooth plane curves
Solving the Dirichlet acoustic scattering problem for a surface with added bumps using the Green's function for the original surface
We solve the Dirichlet problem for acoustic scattering from a
surface which has been perturbed by the addition of one or more
bumps. We build the solution for the bumpy case using the Green's
function for the unperturbed surface, and the solution of a local
integral equation in which the integration is carried out only
over the added bumps. We conclude by giving an alternative
formulation of our method for the special case of a bump on a
plane
Applications of some formulas for finite Markov chains
AbstractWe present some explicit matrix formulas for a finite state Markov chain. The first gives sums of probabilities along some general subsets of paths. Another formula yields the probability mass function (pmf) of the random variable which adds costs along subsets of paths. We then discuss how these formulas can be used to efficiently compute expected values of a function of the sum of costs along paths, as well as related applications. We conclude by describing a procedure allowing us to avoid using Monte Carlo simulation in stochastic approaches to solving some general boundary value problems. Instead, we show how to evaluate the relevant expected values exactly for discretizations of the original continuous problem
A fast numerical method for evaluation of Calderón commutators
AbstractWe describe a methodology for fast evaluation of multilinear operators that are generated by a rapidly computable nonlinear operator. We illustrate this idea by developing a simple numerical algorithm for the fast evaluation of Calderón commutators of all orders,Cnf(x)=p.v.∫−∞∞(A(x)−A(y))n(x−y)n+1f(y)dy,n=1,2,…. The method is based on a representation of the commutators as derivatives of a one parameter family of real-valued versions of Cauchy integrals. We include numerical experiments for the first two commutators. Additionally, we consider the Dirichlet problem for the Laplacian in the unbounded region above the graph of a function. We demonstrate that Calderón commutators appear as building blocks of the functional coefficients of a perturbative solution for this problem