310 research outputs found
Critical points of inner functions, nonlinear partial differential equations, and an extension of Liouville's theorem
We establish an extension of Liouville's classical representation theorem for
solutions of the partial differential equation and combine
this result with methods from nonlinear elliptic PDE to construct holomorphic
maps with prescribed critical points and specified boundary behaviour. For
instance, we show that for every Blaschke sequence in the unit disk
there is always a Blaschke product with as its set of critical
points. Our work is closely related to the Berger-Nirenberg problem in
differential geometry.Comment: 21 page
Variational Integrators for Reduced Magnetohydrodynamics
Reduced magnetohydrodynamics is a simplified set of magnetohydrodynamics
equations with applications to both fusion and astrophysical plasmas,
possessing a noncanonical Hamiltonian structure and consequently a number of
conserved functionals. We propose a new discretisation strategy for these
equations based on a discrete variational principle applied to a formal
Lagrangian. The resulting integrator preserves important quantities like the
total energy, magnetic helicity and cross helicity exactly (up to machine
precision). As the integrator is free of numerical resistivity, spurious
reconnection along current sheets is absent in the ideal case. If effects of
electron inertia are added, reconnection of magnetic field lines is allowed,
although the resulting model still possesses a noncanonical Hamiltonian
structure. After reviewing the conservation laws of the model equations, the
adopted variational principle with the related conservation laws are described
both at the continuous and discrete level. We verify the favourable properties
of the variational integrator in particular with respect to the preservation of
the invariants of the models under consideration and compare with results from
the literature and those of a pseudo-spectral code.Comment: 35 page
Critical sets of bounded analytic functions, zero sets of Bergman spaces and nonpositive curvature
A classical result due to Blaschke states that for every analytic self-map
of the open unit disk of the complex plane there exists a Blaschke product
such that the zero sets of and agree. In this paper we show that
there is an analogue statement for critical sets, i.e. for every analytic
self-map of the open unit disk there is even an indestructible Blaschke
product such that the critical sets of and coincide. We further
relate the problem of describing the critical sets of bounded analytic
functions to the problem of characterizing the zero sets of some weighted
Bergman space as well as to the Berger-Nirenberg problem from differential
geometry. By solving the Berger-Nirenberg problem for a special case we
identify the critical sets of bounded analytic functions with the zero sets of
the weighted Bergman space
Heterogeneous Catalysts for Biodiesel Production
The environmental benefits of biodiesel produced from fully renewable resources should not be underestimated; its production follows the principles of green chemistry and it closes the carbon cycle. Nevertheless, high production costs preclude the replacement of petroleum-based diesel fuel with biodiesel. The introduction of a solid heterogeneous catalyst in biodiesel production contributed to reducing the biodiesel price, thus moving toward competitiveness with diesel. Lin\u27s group was at the forefront of this endeavor and invented and developed one of the first heterogeneous catalysts for generating biodiesel. Nowadays, current catalysts catalyze transesterification and esterification at low temperatures, are stable, especially in water, and have high selectivities
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