Formal conserved quantities for isothermic surfaces

Isothermic surfaces in $S^n$ are characterised by the existence of a pencil $\nabla^t$ of flat connections. Such a surface is special of type $d$ if there is a family $p(t)$ of $\nabla^t$-parallel sections whose dependence on the spectral parameter $t$ is polynomial of degree $d$. We prove that any isothermic surface admits a family of $\nabla^t$-parallel sections which is a formal Laurent series in $t$. As an application, we give conformally invariant conditions for an isothermic surface in $S^3$ to be special.Comment: 13 page

Automated verification of shape and size properties via separation logic.

Despite their popularity and importance, pointer-based programs remain a major challenge for program verification. In this paper, we propose an automated verification system that is concise, precise and expressive for ensuring the safety of pointer-based programs. Our approach uses user-definable shape predicates to allow programmers to describe a wide range of data structures with their associated size properties. To support automatic verification, we design a new entailment checking procedure that can handle well-founded inductive predicates using unfold/fold reasoning. We have proven the soundness and termination of our verification system, and have built a prototype system

Constructing solutions to the Bj\"orling problem for isothermic surfaces by structure preserving discretization

In this article, we study an analog of the Bj\"orling problem for isothermic surfaces (that are more general than minimal surfaces): given a real analytic curve $\gamma$ in ${\mathbb R}^3$, and two analytic non-vanishing orthogonal vector fields $v$ and $w$ along $\gamma$, find an isothermic surface that is tangent to $\gamma$ and that has $v$ and $w$ as principal directions of curvature. We prove that solutions to that problem can be obtained by constructing a family of discrete isothermic surfaces (in the sense of Bobenko and Pinkall) from data that is sampled along $\gamma$, and passing to the limit of vanishing mesh size. The proof relies on a rephrasing of the Gauss-Codazzi-system as analytic Cauchy problem and an in-depth-analysis of its discretization which is induced from the geometry of discrete isothermic surfaces. The discrete-to-continuous limit is carried out for the Christoffel and the Darboux transformations as well.Comment: 29 pages, some figure

Willmore Surfaces of Constant Moebius Curvature

We study Willmore surfaces of constant Moebius curvature $K$ in $S^4$. It is proved that such a surface in $S^3$ must be part of a minimal surface in $R^3$ or the Clifford torus. Another result in this paper is that an isotropic surface (hence also Willmore) in $S^4$ of constant $K$ could only be part of a complex curve in $C^2\cong R^4$ or the Veronese 2-sphere in $S^4$. It is conjectured that they are the only examples possible. The main ingredients of the proofs are over-determined systems and isoparametric functions.Comment: 16 pages. Mistakes occured in the proof to the main theorem (Thm 3.6) has been correcte

A factorization of a super-conformal map

A super-conformal map and a minimal surface are factored into a product of two maps by modeling the Euclidean four-space and the complex Euclidean plane on the set of all quaternions. One of these two maps is a holomorphic map or a meromorphic map. These conformal maps adopt properties of a holomorphic function or a meromorphic function. Analogs of the Liouville theorem, the Schwarz lemma, the Schwarz-Pick theorem, the Weierstrass factorization theorem, the Abel-Jacobi theorem, and a relation between zeros of a minimal surface and branch points of a super-conformal map are obtained.Comment: 21 page

Localized induction equation and pseudospherical surfaces

We describe a close connection between the localized induction equation hierarchy of integrable evolution equations on space curves, and surfaces of constant negative Gauss curvature.Comment: 21 pages, AMSTeX file. To appear in Journal of Physics A: Mathematical and Genera

A reification calculus for model-oriented software specification

This paper presents a transformational approach to the derivation of implementations from model-oriented specifications of abstract data types. The purpose of this research is to reduce the number of formal proofs required in model refinement, which hinder software development. It is shown to be appli- cable to the transformation of models written in Meta-iv (the specification lan- guage of Vdm) towards their refinement into, for example, Pascal or relational DBMSs. The approach includes the automatic synthesis of retrieve functions between models, and data-type invariants. The underlying algebraic semantics is the so-called final semantics “`a la Wand”: a specification “is” a model (heterogeneous algebra) which is the final ob ject (up to isomorphism) in the category of all its implementations. The transformational calculus approached in this paper follows from exploring the properties of finite, recursively defined sets. This work extends the well-known strategy of program transformation to model transformation, adding to previous work on a transformational style for operation- decomposition in META-IV. The model-calculus is also useful for improving model-oriented specifications.(undefined