Spectrum of the Magnetic Schrödinger Operator in a Waveguide with Combined Boundary Conditions

We consider the magnetic Schrödinger operator in a two-dimensional strip. On the boundary of the strip the Dirichlet boundary condition is imposed except for a fixed segment (window), where it switches to magnetic Neumann {For the definition of magnetic Neumann boundary conditions see Section 2, Eq. (2.2)}. We deal with a smooth compactly supported field as well as with the Aharonov-Bohm field. We give an estimate on the maximal length of the window, for which the discrete spectrum of the considered operator will be empty. In the case of a compactly supported field we also give a sufficient condition for the presence of eigenvalues below the essential spectrum.submitted 11/05/04, accepted 21/09/04     

Visualizing elements of order two in the Weil-Châtelet group

Let E be an elliptic curve over an infinite field K with characteristic ≠2, and σ∈H¹(G_K,E)[2] a two-torsion element of its Weil-Châtelet group. We prove that σ is always visible in infinitely many abelian surfaces up to isomorphism, in the sense put forward by Cremona and Mazur in their article (J. Exp. Math. 9(1) (2000) 13). Our argument is a variant of Mazur's proof, given in (Asian J. Math. 3(1) (1999) 221), for the analogous statement about three-torsion elements of the Shafarevich-Tate group in the setting where K is a number field. In particular, instead of the universal elliptic curve with full level-three-structure, our proof makes use of the universal elliptic curve with full level-two-structure and an invariant differential.     

Introducing cost-plus-loss analysis into a hierarchical forestry planning environment

Cost-plus-loss analysis of data for forestry planning has often been carried out for highly simplified planning situations. In this study, we suggest an advance in the cost-plus-loss methodology that aims to capture the hierarchical structure and iterative nature of planning by the large forest owner. The simulation system that is developed to simulate the planning process of the forest owner includes the tactical and operational levels of a continuous planning process. The system is characterized by annual re-planning of the tactical plan with a planning horizon of ten year and with the option to reassess data for selected stands before operational planning. Operational planning is done with a planning horizon of two years and the first year of the plan is considered to have been executed before moving the planning process one year forward. The annual cycle is repeated 10 times, simulating decisions made over a ten-year time horizon. The optimizing planning models of the system consider wood flow requirements, available harvest resources, seasonal variation of ground conditions and spatiality. The data used are evaluated according to standard procedures in cost-plus-loss analysis. Results from a test case indicate high decision losses when planning at both levels is based on the type of data prevalent in the stand databases of Swedish companies. The losses can be reduced substantially if higher-quality data are introduced before operational planning. In summary, the results indicate that the method makes it possible to analyze where in the planning process one needs better data and their value.     

A variational conjugate gradient method for determining the fluid velocity of a slow viscous flow

The problem considered is that of determining the fluid velocity for linear hydrostatics Stokes flow of slow viscous fluids from measured velocity and fluid stress force on a part of the boundary of a bounded domain. A variational conjugate gradient iterative procedure is proposed based on solving a series of mixed well-posed boundary value problems for the Stokes operator and its adjoint. In order to stabilize the Cauchy problem, the iterations are ceased according to an optimal order discrepancy principle stopping criterion. Numerical results obtained using the boundary element method confirm that the procedure produces a convergent and stable numerical solution.     

Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions

We consider the distribution of the orbits of the number 1 under the $\beta $ -transformations $T_\beta $ as $\beta $ varies. Mainly, the size of the set of $\beta >1$ for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension. The dimension of the following set $E\big (\{\ell _n\}_{n\ge 1}, x_0\big )=\Big \{\,\beta >1: |T^n_{\beta }1-x_0|<\beta ^{-\ell _n}, \hbox { for infinitely many}, \, n\in \mathbb{N }\,\Big \}$ is determined, where $x_0$ is a given point in $[0,1]$ and $\{\ell _n\}_{n\ge 1}$ is a sequence of integers tending to infinity as $n\rightarrow \infty $ . For the proof of this result, the notion of the recurrence time of a word in symbolic space is introduced to characterise the lengths and the distribution of cylinders (the set of $\beta $ with a common prefix in the expansion of 1) in the parameter space $\{\,\beta \in \mathbb{R }: \beta >1\,\}$ .     

Galois theory and commutators

We prove that the relative commutator with respect to a subvariety of a variety of Ω-groups introduced by the first author can be described in terms of categorical Galois theory. This extends the known correspondence between the Fröhlich–Lue and the Janelidze–Kelly notions of central extension. As an example outside the context of Ω-groups we study the reflection of the category of loops to the category of groups where we obtain an interpretation of the associator as a relative commutator.     

Does Huygens' principle hold for small perturbations of the wave equation?

The perturbed wave equation □u + q(x)u = 0 in $\text{R}$³ × $\text{R}$ with C∞ ($\text{R}$³) compactly supported initial data at t = 0 is considered. It is proven that the Huygens' principle does not hold for this equation if the potential is (essentially) non-negative, well-behaved at infinity and small in a suitable sense. The treatment is elementary and based on energy estimates and the positivity of the Riemann function for the wave equation in three space dimensions. The result still holds if the solution u is “small” over some space-time propagation cone. In the ease in which q has compact support, stronger results of this type for the above equation are obtained.