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We analyze the global structure of Lagrangian Grassmannian in the product symplectic space and investigate the local properties of generic symplectic relations. The cohomological symplectic invariant of discrete dynamical systems is generalized to the class of generalized canonical mappings. Lower bounds for the number of two-point and three-point symplectic invariants for billiard-type dynamical systems are found and several examples of symplectic correspondences encountered from physics are presented. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Comments on several articles related to the field of physics as of 1981. Deterioration in both quality and quantity of secondary physics education in the U.S.; Problem of developing a constructive dialog between librarians and publishers to solve a photocopying problem; Morality of people who have legitimate concerns about the safety of nuclear power plants.

The helicity of a smooth vector field defined on a domain in three-space is the standard measure of the extent to which the field lines wrap and coil around one another. It plays important roles in fluid mechanics, magnetohydrodynamics, and plasma physics. The isoperimetric problem in this setting is to maximize helicity among all divergence-free vector fields of given energy, defined on and tangent to the boundary of all domains of given volume in three-space. The Biot-Savart operator starts with a divergence-free vector field defined on and tangent to the boundary of a domain in three-space, regards it as a distribution of electric current, and computes its magnetic field. Restricting the magnetic field to the given domain, we modify it by subtracting a gradient vector field so as to keep it divergence-free while making it tangent to the boundary of the domain. The resulting operator, when extended to the L[sup 2] completion of this family of vector fields, is compact and self-adjoint, and thus has a largest eigenvalue, whose corresponding eigenfields are smooth by elliptic regularity. The isoperimetric problem for this modified Biot-Savart operator is to maximize its largest eigenvalue among all domains of given volume in three-space. The curl operator, when restricted to the image of the modified Biot-Savart operator, is its inverse, and the isoperimetric problem for this restriction of the curl is to minimize its smallest positive eigenvalue among all domains of given volume in three-space. These three isoperimetric problems are equivalent to one another. In this paper, we will derive the first variation formulas appropriate to these problems, and use them to constrain the nature of any possible solution. For example, suppose that the vector field V, defined on the compact, smoothly bounded domain Ω, maximizes helicity among all divergence-free vector fields of given nonzero energy, defined on and tangent to the boundary of all such domains of g... [ABSTRACT FROM AUTHOR]

The basic spin difference character Δ[sup ″] of SO(2n) is a useful device in dealing with characters of irreducible spinor representations of SO(2n). It is shown here that its kth-fold symmetrized powers, or plethysms, associated with partitions κ of k factorize in such a way that Δ[sup ″]x{κ}=(Δ[sup ″])[sup r(κ)]Π[sub κ], where r(κ) is the Frobenius rank of κ. The analogy between SO(2n) and Sp(2n,R) is shown to be such that the plethysms of the basic harmonic or metaplectic character Δ˜ of Sp(2n,R) factorize in the same way to give Δ˜x{κ}=(Δ˜)[sup r(κ)]Π˜[sub κ]. Moreover, the analogy is shown to extend to the explicit decompositions into characters of irreducible representations of SO(2n) and Sp(2n,R) not only for the plethysms themselves, but also for their factors Π[sub κ] and Π˜[sub κ]. Explicit formulas are derived for each of these decompositions, expressed in terms of various group-subgroup branching rule multiplicities, particularly those defined by the restriction from O(k) to the symmetric group S[sub k]. Illustrative examples are included, as well as an extension to the symmetrized powers of certain basic tensor difference characters of both SO(2n) and Sp(2n,R). © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The analogy between the finite-dimensional spin representation Δ of SO(2n) and the infinite-dimensional representation Δ˜ of Sp(2n,R) is made precise. It is then shown that this analogy can be extended so as to provide a precise link between each finite dimensional unitary irreducible representation of SO(2n) and a corresponding infinite-dimensional unitary irreducible representation of Sp(2n,R). The analogy shows itself at the level of the corresponding characters and difference characters, and involves the use of Schur function methods to express both characters and difference characters of SO(2n) and Sp(2n,R) in terms of characters of irreducible representations of their common subgroup U(n). The analogy is extended still further to cover the explicit decomposition of not only tensor products of Δ and Δ˜ with other unitary irreducible representations of SO(2n) and Sp(2n,R), respectively, but also arbitrary tensor powers of Δ and Δ˜. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]