New constructions of twistor lifts for harmonic maps

We show that given a harmonic map $\varphi$ from a Riemann surface to a classical compact simply connected inner symmetric space, there is a $J_2$-holomorphic twistor lift of $\varphi$ (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae in terms of holomorphic data which describes their extended solutions. In particular, this gives explicit formulae for the twistor lifts of all harmonic maps of finite uniton number from a surface to the above symmetric spaces.Comment: Some minor changes and a correction of Example 8.

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

Imperative Object-based Calculi in (Co)Inductive Type Theories

We discuss the formalization of Abadi and Cardelli's imps, a paradigmatic object-based calculus with types and side effects, in Co-Inductive Type Theories, such as the Calculus of (Co)Inductive Constructions (CC(Co)Ind). Instead of representing directly the original system "as it is", we reformulate its syntax and semantics bearing in mind the proof-theoretical features provided by the target metalanguage. On one hand, this methodology allows for a smoother implementation and treatment of the calculus in the metalanguage. On the other, it is possible to see the calculus from a new perspective, thus having the occasion to suggest original and cleaner presentations. We give hence anew presentation of imps, exploiting natural deduction semantics, (weak) higher-order abstract syntax, and, for a significant fragment of the calculus, coinductive typing systems. This presentation is easier to use and implement than the original one, and the proofs of key metaproperties, e.g. subject reduction, are much simpler. Although all proof developments have been carried out in the Coq system, the solutions we have devised in the encoding of and metareasoning on imps can be applied to other imperative calculi and proof environments with similar features

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 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

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

Hybridization of institutions

Extended version including all proofsModal logics are successfully used as specification logics for reactive systems. However, they are not expressive enough to refer to individual states and reason about the local behaviour of such systems. This limitation is overcome in hybrid logics which introduce special symbols for naming states in models. Actually, hybrid logics have recently regained interest, resulting in a number of new results and techniques as well as applications to software specification. In this context, the first contribution of this paper is an attempt to ‘universalize’ the hybridization idea. Following the lines of [DS07], where a method to modalize arbitrary institutions is presented, the paper introduces a method to hybridize logics at the same institution-independent level. The method extends arbitrary institutions with Kripke semantics (for multi-modalities with arbitrary arities) and hybrid features. This paves the ground for a general result: any encoding (expressed as comorphism) from an arbitrary institution to first order logic (FOL) deter- mines a comorphism from its hybridization to FOL. This second contribution opens the possibility of effective tool support to specification languages based upon logics with hybrid features.Fundação para a Ciência e a Tecnologia (FCT