Parametric polynomial minimal surfaces of arbitrary degree

Weierstrass representation is a classical parameterization of minimal surfaces. However, two functions should be specified to construct the parametric form in Weierestrass representation. In this paper, we propose an explicit parametric form for a class of parametric polynomial minimal surfaces of arbitrary degree. It includes the classical Enneper surface for cubic case. The proposed minimal surfaces also have some interesting properties such as symmetry, containing straight lines and self-intersections. According to the shape properties, the proposed minimal surface can be classified into four categories with respect to $n=4k-1$ $n=4k+1$, $n=4k$ and $n=4k+2$. The explicit parametric form of corresponding conjugate minimal surfaces is given and the isometric deformation is also implemented

Bi-quartic parametric polynomial minimal surfaces

Minimal surfaces with isothermal parameters admitting B\'{e}zier representation were studied by Cosin and Monterde. They showed that, up to an affine transformation, the Enneper surface is the only bi-cubic isothermal minimal surface. Here we study bi-quartic isothermal minimal surfaces and establish the general form of their generating functions in the Weierstrass representation formula. We apply an approach proposed by Ganchev to compute the normal curvature and show that, in contrast to the bi-cubic case, there is a variety of bi-quartic isothermal minimal surfaces. Based on the Bezier representation we establish some geometric properties of the bi-quartic harmonic surfaces. Numerical experiments are visualized and presented to illustrate and support our results.Comment: 14 page

Numerical Simulations of Directed Self-Assembly Methods in Di-block Copolymer Films for Efficient Manufacturing of Nanoscale Patterns with Long-Range Order

Directed self-assembly (DSA) of block copolymers (BCPs) has been shown as a viable method to achieve bulk fabrication of surface patterns with feature sizes smaller than those available through traditional photolithography. Under appropriate thermodynamic conditions, BCPs will self-assemble into ordered micro-domain morphologies, a desirable feature for many applications. One of the primary interests in this field of research is the application of thin-film BCPs to existing photolithography techniques. This “bottom-up” approach utilizes the self-assembled BCP nanostructures as a sacrificial templating layer in the lithographic process. While self-assembly occurs spontaneously, extending orientational uniformity over centimeter-length scales remains a critical challenge. A number of DSA techniques have been developed to enhance the long range order in an evolving BCP system during micro-phase separation. Of primary interest to this dissertation is the synergistic behavior between chemoepitaxial templating and cold-zone annealing. The first method involves pre-treating a substrate with chemical boundaries that will attract or repel one of the monomer blocks before application of the thin-film via spin-coating. The second method applies a mobile, thermal gradient to induce micro-phase separation in a narrow region within the homogeneous thin-film . Parametric studies have been performed to characterize the extent of long range order and defect densities obtained by applying various thermal zone velocities and template patterns. These simulations are performed by utilizing a Time-Dependent Ginzburg-Landau (TDGL) model and an optimized phase field (OPF) model. Parallel processing is implemented to allow large-scale simulations to be performed within a reasonable time period

Complexity of Model Testing for Dynamical Systems with Toric Steady States

In this paper we investigate the complexity of model selection and model testing for dynamical systems with toric steady states. Such systems frequently arise in the study of chemical reaction networks. We do this by formulating these tasks as a constrained optimization problem in Euclidean space. This optimization problem is known as a Euclidean distance problem; the complexity of solving this problem is measured by an invariant called the Euclidean distance (ED) degree. We determine closed-form expressions for the ED degree of the steady states of several families of chemical reaction networks with toric steady states and arbitrarily many reactions. To illustrate the utility of this work we show how the ED degree can be used as a tool for estimating the computational cost of solving the model testing and model selection problems