217 research outputs found
On the combinatorics of Demoulin transforms and (discrete) projective minimal surfaces
The classical Demoulin transformation is examined in the context of discrete differential geometry. We show that iterative application of the Demoulin transformation to a seed projective minimal surface generates a Z 2 lattice of projective minimal surfaces. Known and novel geometric properties of these Demoulin lattices are discussed and used to motivate the notion of lattice Lie quadrics and associated discrete envelopes and the definition of the class of discrete projective minimal and Q-surfaces (PMQ-surfaces). We demonstrate that the even and odd Demoulin sublattices encode a two-parameter family of pairs of discrete PMQ-surfaces with the property that one discrete PMQ-surface constitute an envelope of the lattice Lie quadrics associated with the other
Surface theory in discrete projective differential geometry. I. A canonical frame and an integrable discrete Demoulin system
We present the first steps of a procedure which discretises surface theory in
classical projective differential geometry in such a manner that underlying
integrable structure is preserved. We propose a canonical frame in terms of
which the associated projective Gauss-Weingarten and Gauss-Mainardi-Codazzi
equations adopt compact forms. Based on a scaling symmetry which injects a
parameter into the linear Gauss-Weingarten equations, we set down an algebraic
classification scheme of discrete projective minimal surfaces which turns out
to admit a geometric counterpart formulated in terms of discrete notions of Lie
quadrics and their envelopes. In the case of discrete Demoulin surfaces, we
derive a Backlund transformation for the underlying discrete Demoulin system
and show how the latter may be formulated as a two-component generalisation of
the integrable discrete Tzitzeica equation which has originally been derived in
a different context. At the geometric level, this connection leads to the
retrieval of the standard discretisation of affine spheres in affine
differential geometry
Two-dimensional protein crystallization via metal-ion coordination by naturally occurring surface histidines
A powerful and potentially general approach to the targeting and crystallization of proteins on lipid interfaces through coordination of surface histidine residues to lipid-chelated divalent metal ions is presented. This approach, which should be applicable to the crystallization of a wide range of naturally occurring or engineered proteins, is illustrated here by the crystallization of streptavidin on a monolayer of an iminodiacetate-Cu(II) lipid spread at the air-water interface. This method allows control of the protein orientation at interfaces, which is significant for the facile production of highly ordered protein arrays and for electron density mapping in structural analysis of two-dimensional crystals. Binding of native streptavidin to the iminodiacetate-Cu lipids occurs via His-87, located on the protein surface near the biotin binding pocket. The two-dimensional streptavidin crystals show a previously undescribed microscopic shape that differs from that of crystals formed beneath biotinylated lipids
On the linearization of the generalized Ermakov systems
A linearization procedure is proposed for Ermakov systems with frequency
depending on dynamic variables. The procedure applies to a wide class of
generalized Ermakov systems which are linearizable in a manner similar to that
applicable to usual Ermakov systems. The Kepler--Ermakov systems belong into
this category but others, more generic, systems are also included
Tzitz\'eica transformation is a dressing action
We classify the simplest rational elements in a twisted loop group, and prove
that dressing actions of them on proper indefinite affine spheres give the
classical Tzitz\'eica transformation and its dual. We also give the group point
of view of the Permutability Theorem, construct complex Tzitz\'eica
transformations, and discuss the group structure for these transformations
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