Beam on a Nonlinear Winkler Foundation (BNWF) approach is widely used in practice since the simulation process is rather simple and requires less computational efforts compared to three-dimensional continuum modeling approaches. In the recent decades, this approach has been extended by proposing advanced onedimensional models in order to better simulate the soil-pile interaction. In this paper,...

An explicit construction of pairing-friendly hyperelliptic curves with ordinary Jacobians was firstly given by D. Freeman. In this paper, we give other explicit constructions of pairing-friendly hyperelliptic curves with ordinary Jacobians based on the closed formulae for the order of the Jacobian of a hyperelliptic curve of type y = x + ax. We present two methods in this paper. One is an analo...

In contrast to the mid point technique, which may not be successful in a region where two edges of a curve cross each other, this paper introduces a unified approach called Method of Deviation that draws digital curve transforming the implicit equation of analog curve into algorithm. The method involves finding the deviation of the location of the pixel from the location of the corresponding po...

Starting with a given planar cubic curve [x(t), y(t)]T, we construct Pythagorean hodograph (PH) space curves of the form [x(t), y(t), z(t)]T in Euclidean and in Minkowski space, which interpolate the tangent vector at a given point. We prove the existence of these curves for any regular planar cubic and we express all solutions explicitly. It is shown that the constructed curves provide upper a...

The group generated by the Weierstrass points of a smooth curve in its Jacobian is an intrinsic invariant of the curve. We determine this group for all smooth quartics with eight hyperflexes or more. Since Weierstrass points are closely related to moduli spaces of curves, as an application, we get bounds on both the rank and the torsion part of this group for a generic quartic having a fixed nu...

ACKNOWLEDGEMENTS I would like to thank my adviser Michael Zieve for teaching me to fish. I would like to thank Jeffrey Lagarias for helpful feedback in writing this thesis. I would like to thank my family (Peter, Susan, and Holly), my beautiful girlfriend Becky, and the community of Michigan graduate students for keeping me sane.

[1] The strong jet, with a speed between 500 and 600 m/s, is inferred in the equatorial region of Saturn by combining the nadir and limb observations of Composite Infrared Spectrometer (CIRS) aboard the Cassini spacecraft. A similar jet was discovered on Jupiter (F. M. Flasar et al., 2004a). These discoveries raise the possibility that intense jets are common in the equatorial stratospheres of ...

We characterize the algebraic invariant curves for the Riccati polynomial differential systems of the form x′ = 1, y′ = a(x)y+ b(x)y+ c(x), where a(x), b(x) and c(x) are arbitrary polynomials. We also characterize their algebraic first integrals.

We consider an Abel equation (∗)y′ = p(x)y2+q(x)y3 withp(x), q(x)—polynomials in x. A centre condition for this equation (closely related to the classical centre condition for polynomial vector fields on the plane) is that y0 = y(0) ≡ y(1) for any solution y(x). This condition is given by the vanishing of all the Taylor coefficients vk(1) in the development y(x) = y0 + ∑∞ k=2 vk(x)y k 0 . Follo...

In this paper, we consider the norm-trace curves which are defined by the equation y r−1 + y r−2 + · · ·+ y = x qr−1 q−1 over IFqr where q is a power of a prime number and r ≥ 2 is an integer. We determine the Weierstrass semigroup of the triple of points (P∞, P00, P0b) on this curve.