Hamiltonian formalisms for multidimensional calculus of variations and perturbation theory

In a first part we propose an introduction to multisymplectic formalisms, which are generalisations of Hamilton's formulation of Mechanics to the calculus of variations with several variables: we give some physical motivations, related to the quantum field theory, and expound the simplest example, based on a theory due to T. de Donder and H. Weyl. In a second part we explain quickly a work in collaboration with J. Kouneiher (math-ph/0211046) on generalizations of the de Donder--Weyl theory (known as Lepage theories). Lastly we show that in this framework a perturbative classical field theory (analog of the perturbative quantum field theory) can be constructed.Comment: 24 pages, conference in the honour of H. Brezis and F. Browder, Rutgers University, October 200

Hamiltonian stationary Lagrangian surfaces in Hermitian symmetric spaces

This paper is the third of a series on Hamiltonian stationary Lagrangian surfaces. We present here the most general theory, valid for any Hermitian symmetric target space. Using well-chosen moving frame formalism, we show that the equations are equivalent to an integrable system, generalizing the C^2 subcase analyzed in the first article (arXiv:math.DG/0009202). This system shares many features with the harmonic map equation of surfaces into symmetric spaces, allowing us to develop a theory close to Dorfmeister, Pedit and Wu's, including for instance a Weierstrass-type representation. Notice that this article encompasses the article mentioned above, although much fewer details will be given on that particular flat case

A Church Based Curriculum Providing Marriage and Family Education for Stepfamilies and Ministries

Within the American population, the Stepfamily now outnumbers that of the traditional nuclear family; a trend mirrored in the local church. Stepfamilies within the church culture struggle in navigating the relational complexities, emotional and spiritual strain, and biblical confusion inherent to the formation of a stepfamily, while church leaders lack both understanding and the training necessary to minister to this growing population. Often step-families choose anonymity or leave the local church for fear of being shunned. In synthesizing, existing research with data garnered from stepfamilies via their participation in an anonymous online survey and participatory learning activity (PLA), challenges unique to stepfamily members in relation to the church can be classified. Armed with this data, a comprehensive and biblically based curriculum will be developed to train members of stepfamilies and those who minister to them

Description of surfaces associated with CPN−1CP^{N-1} sigma models on Minkowski space

The objective of this paper is to construct and investigate smooth orientable surfaces in $R^{N^2-1}$ by analytical methods. The structural equations of surfaces in connection with $CP^{N-1}$ sigma models on Minkowski space are studied in detail. This is carried out using moving frames adapted to surfaces immersed in the $su(N)$ algebra. The first and second fundamental forms of this surface as well as the relations between them as expressed in the Gauss-Weingarten and Gauss-Codazzi-Ricci equations are found. The Gaussian curvature, the mean curvature vector and the Willmore functional expressed in terms of a solution of $CP^{N-1}$ sigma model are obtained. An example of a surface associated with the $CP^1$ model is included as an illustration of the theoretical results.Comment: 19 pages, 1 figure; shorter version, some typos and minor mistakes correcte

Detariffing and the Death of the Filed Tariff Doctrine: Deregulating in the “Self” Interest

This Article reviews the history of the FCC\u27s detariffing efforts, addressing the major issue raised not so much by detariffing itself, but by the FCC\u27s view of detariffing orders impact on the Filed Tariff Doctrine. Notwithstanding the existence of the Doctrine for nearly a century, the FCC, through detariffing, has declared the Doctrine dead. This Article formally opposes the FCC\u27s declaration and suggests that the FCC\u27s motivations behind detariffing have failed to consider, much less attempted to properly balance, the conflicting public interests involved. Comparing and contrasting the legal rights enjoyed by long-distance carriers under the Filed Tariff Doctrine to the rights and potential liabilities of carriers in its absence, this Article discusses the actions several states have taken immediately following the July 31, 2001 effective date of the FCC\u27s mass-market detariffing order. The Article concludes that the FCC\u27s action is a manifest injustice to both carriers and consumers alike, and is a prime example of irresponsible agency regulation

Conformally parametrized surfaces associated with CP^(N-1) sigma models

Two-dimensional conformally parametrized surfaces immersed in the su(N) algebra are investigated. The focus is on surfaces parametrized by solutions of the equations for the CP^(N-1) sigma model. The Lie-point symmetries of the CP^(N-1) model are computed for arbitrary N. The Weierstrass formula for immersion is determined and an explicit formula for a moving frame on a surface is constructed. This allows us to determine the structural equations and geometrical properties of surfaces in R^(N^2-1). The fundamental forms, Gaussian and mean curvatures, Willmore functional and topological charge of surfaces are given explicitly in terms of any holomorphic solution of the CP^2 model. The approach is illustrated through several examples, including surfaces immersed in low-dimensional su(N) algebras.Comment: 32 page