Action on flag varieties: 2-Dimensional case

LetA be a finite dimensional commutative semisimple algebra over a fieldk and letV be a finitely generatedA-module. We examine the action of the general linear group GL _A (V) on the set of flags ofk-subspaces ofV. Also, let (V, B) be a finitely generated symplectic module overA. We also investigate the action of the symplectic group Sp _A (V, B) on the set of flags ofB-isotropick-subspaces ofV, whereB=°B is thek-symplectic form induced by a nonzerok-linear map :A k. In both cases, the orbits are completely classified in terms of certain integer invariants provided that dim _{k A}=2.This work is partially supported by a KOSEF research grant.     

On centroid branches of trees from certain families

We obtain various results involving the centroid branches of randomly chosen trees from simply generated families.     

A Schur-Weyl Duality Approach toWalking on Cubes

Walks on the representation graph \({\mathcal{R}_{\mathsf{V}}}\)(G) determined by a group G and a G-module V are related to the centralizer algebras of the action of G on the tensor powers \({\mathsf{V}^{\otimes k}}\) via Schur-Weyl duality. This paper explores that connection when the group is \({\mathbb{Z}^{n}_{2}}\) and the module V is chosen so the representation graph is the n-cube. We describe a basis for the centralizer algebras in terms of labeled partition diagrams. We obtain an expression for the number of walks by counting certain partitions and determine the exponential generating functions for the number of walks.     

The preparation of 2,6-diaminopyrazine, 2,6-diazidopyrazine and some of their derivatives

A much improved synthesis of the heretofore difficultly obtainable 2,6-diaminopyrazine (4) was afforded by the low-pressure catalytic hydrogenation (palladium on carbon) of 2,6-diazido-pyrazine (2) ; reaction of 2,6-dichloropyrazine (1) and sodium azide gave 2 in 84% yield. The outcome of the reduction was found to be solvent dependent: 1,2-dimethoxyethane containing aqueous ammonia gave 4 in 83% yield; 1,2-dimethoxyethane alone gave 5-aminotetrazolo[1,5-a]-pyrazine (3) in 26% yield. Additional alternative syntheses of 3 and 4 are described. A number of acyl and azo derivatives of 4 were prepared. Reactions of 2 with dimethyl acetylenedicarboxylate and ethyl acetate (base catalyzed) leading to vic-triazole derivatives are also described.     

A multilevel finite element method in space‐time for the Navier‐Stokes problem

A multilevel finite element method in space‐time for the two‐dimensional nonstationary Navier‐Stokes problem is considered. The method is a multi‐scale method in which the fully nonlinear Navier‐Stokes problem is only solved on a single coarsest space‐time mesh; subsequent approximations are generated on a succession of refined space‐time meshes by solving a linearized Navier‐Stokes problem about the solution on the previous level. The a priori estimates and error analysis are also presented for the J‐level finite element method. We demonstrate theoretically that for an appropriate choice of space and time mesh widths: h_j ～ h, k_j ～ k, j = 2, …, J, the J‐level finite element method in space‐time provides the same accuracy as the one‐level method in space‐time in which the fully nonlinear Navier‐Stokes problem is solved on a final finest space‐time mesh. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Liouville type theorems for p-harmonic maps

Let M be a complete Riemannian manifold and let N be a Riemannian manifold of non-positive sectional curvature. Assume that at all x∈M and at some point x₀∈M, where μ₀>0 is the least eigenvalue of the Laplacian acting on L²-functions on M. Let 2?q?p. Then any q-harmonic map of finite q-energy is constant. Moreover, if N is a Riemannian manifold of non-positive scalar curvature, then any q-harmonic morphism of finite q-energy is constant.     

Weighted Hamming metric structures

It is known that the binary generalized Goppa codes are perfect codes for the weighted Hamming metrics. In this paper, we present the existence of a weighted Hamming metric that admits a binary Hamming code (resp. an extended binary Hamming code) to be perfect code. For a special weighted Hamming metric, we also give some structures of a 2-perfect code, show how to construct a 2-perfect linear code and obtain the weight distribution of a 2-perfect code from the partial information of the code.     

Rational frames of minimal twist along space curves under specified boundary conditions

An adapted orthonormal frame (f₁(ξ),f₂(ξ),f₃(ξ)) on a space curve r(ξ), ξ ∈ [ 0, 1 ] comprises the curve tangent \(\mathbf {f}_{1}(\xi ) =\mathbf {r}^{\prime }(\xi )/|\mathbf {r}^{\prime }(\xi )|\) and two unit vectors f₂(ξ),f₃(ξ) that span the normal plane. The variation of this frame is specified by its angular velocity Ω = Ω₁f₁ + Ω₂f₂ + Ω₃f₃, and the twist of the framed curve is the integral of the component Ω₁ with respect to arc length. A minimal twist frame (MTF) has the least possible twist value, subject to prescribed initial and final orientations f₂(0),f₃(0) and f₂(1),f₃(1) of the normal–plane vectors. Employing the Euler–Rodrigues frame (ERF) — a rational adapted frame defined on spatial Pythagorean–hodograph curves — as an intermediary, an exact expression for an MTF with Ω₁ = constant is derived. However, since this involves rather complicated transcendental terms, a construction of rational MTFs is proposed by the imposition of a rational rotation on the ERF normal–plane vectors. For spatial PH quintics, it is shown that rational MTFs compatible with the boundary conditions can be constructed, with only modest deviations of Ω₁ about the mean value, by a rational quartic normal–plane rotation of the ERF. If necessary, subdivision methods can be invoked to ensure that the rational MTF is free of inflections, or to more accurately approximate a constant Ω₁. The procedure is summarized by an algorithm outline, and illustrated by a representative selection of computed examples.