Symmetry, Conserved Charges, and Lax Representations of Nonlinear Field Equations: A Unified Approach

A certain non-Noetherian connection between symmetry and integrability properties of nonlinear field equations in conservation-law form is studied. It is shown that the symmetry condition alone may lead, in a rather straightforward way, to the construction of a Lax pair, a doubly infinite set of (generally nonlocal) conservation laws, and a recursion operator for symmetries. Applications include the chiral field equation and the self-dual Yang-Mills equation.Comment: 15 page

Infinitesimal symmetry transformations of matrix-valued differential equations: An algebraic approach

The study of symmetries of partial differential equations (PDEs) has been traditionally treated as a geometrical problem. Although geometrical methods have been proven effective with regard to finding infinitesimal symmetry transformations, they present certain conceptual difficulties in the case of matrix-valued PDEs; for example, the usual differential-operator representation of the symmetry-generating vector fields is not possible in this case. An algebraic approach to the symmetry problem of PDEs is described, based on abstract operators (characteristic derivatives) which admit a standard differential-operator representation in the case of scalar-valued PDEs.Comment: 17 pages; minor corrections; see also arXiv:0803.368

The Maxwell equations as a B\"acklund transformation

Backlund transformations (BTs) are a useful tool for integrating nonlinear partial differential equations (PDEs). However, the significance of BTs in linear problems should not be ignored. In fact, an important linear system of PDEs in Physics, namely, the Maxwell equations of Electromagnetism, may be viewed as a BT relating the wave equations for the electric and the magnetic field, these equations representing integrability conditions for solution of the Maxwell system. We examine the BT property of this system in detail, both for the vacuum case and for the case of a linear conducting medium.Comment: 7 pages; minor corrections; see also arXiv:0803.3688 and references therei

A hybrid architecture for the implementation of the Athena neural net model

The implementation of an earlier introduced neural net model for pattern classification is considered. Data flow principles are employed in the development of a machine that efficiently implements the model and can be useful for real time classification tasks. Further enhancement with optical computing structures is also considered

Electromotive Force: A Guide for the Perplexed

The concept of electromotive force (emf) may be introduced in various ways in an undergraduate course of theoretical electromagnetism. The multitude of alternate expressions for the emf is often the source of confusion to the student. We summarize the main ideas, adopting a pedagogical logic that proceeds from the general to the specific. The emf of a "circuit" is first defined in the most general terms. The expressions for the emf of some familiar electrodynamical systems are then derived in a rather straightforward manner. A diversity of physical situations is thus unified within a common theoretical framework.Comment: 11 pages, 5 figures; added example & appendi

The Essential Role Of Renewable Energy In A Sustainable Future

This paper examines why renewable energy is essential to building a healthy, reliable, and secure future, that ultimately, will be both more cost efficient and beneficial to the environment. Despite the reluctance of the American national government to commit itself to renewable energy, several countries and California are demonstrating how renewable energy can become a significant part of their economy in less than a decade. While private corporations are important, government incentives and disincentives play a decisive role in the successful conversion to renewables

Some Lessons Learned For Sustainable Development From The Mexican Experience

The economic reforms that are necessary to implement the neo-liberal paradigm have to be tempered in developing countries by the crucial roles of the national government in the organization, administration and monitoring of institutions that will carry out these reforms.  In addition to these reforms, countries like Mexico, where half the population lives below the poverty line, need a nationally-funded and administered program to uplift the rural and urban poor.  Mexico has developed such a program that has significantly improved the health and nutrition of mothers and children and class attendance.  On the other hand, federal governments of wealthy nations, who espouse free market operations, can by their actions thwart free trade in order to win elections.  We examine how very large agricultural subsidies to American farmers and corporations distort the price of corn and corn products in Mexico in a way that makes it difficult for many small farmers to survive. These subsidies go against the free trade principles of the North American Free Trade Agreement (NAFTA), signed by the United States, Mexico and Canada in 1994, but are justified by loopholes included in the final text.  The concept of sustainable development provides a way of looking at development programs from a long-term view, with the welfare of the next generation in mind. While the poverty program, described above, has been shown to improve the nutrition. health and school attendance of many poor children, so they can expect to lead a better life than their parents, the NAFTA free trade treaty has worked to enrich a few multi-national businesses at the expense of the welfare of many farmers throughout Mexico and North America. Having used organic methods of farming that preserved the land's fertility and productivity, Mexican small farmers now have to leave the land for the over-crowded cities with little hope of making a good livelihood for their families

Foundations of Newtonian Dynamics: An Axiomatic Approach for the Thinking Student

Despite its apparent simplicity, Newtonian Mechanics contains conceptual subtleties that may cause some confusion to the deep-thinking student. These subtleties concern fundamental issues such as, e.g., the number of independent laws needed to formulate the theory, or, the distinction between genuine physical laws and derivative theorems. This article attempts to clarify these issues for the benefit of the student by revisiting the foundations of Newtonian Dynamics and by proposing a rigorous axiomatic approach to the subject. This theoretical scheme is built upon two fundamental postulates, namely, conservation of momentum and superposition property for interactions. Newton's Laws, as well as all familiar theorems of Mechanics, are shown to follow from these basic principles.Comment: 11 pages; extensively revised and expanded versio