150 research outputs found
Solutions of the sDiff(2)Toda equation with SU(2) Symmetry
We present the general solution to the Plebanski equation for an H-space that
admits Killing vectors for an entire SU(2) of symmetries, which is therefore
also the general solution of the sDiff(2)Toda equation that allows these
symmetries. Desiring these solutions as a bridge toward the future for yet more
general solutions of the sDiff(2)Toda equation, we generalize the earlier work
of Olivier, on the Atiyah-Hitchin metric, and re-formulate work of Babich and
Korotkin, and Tod, on the Bianchi IX approach to a metric with an SU(2) of
symmetries. We also give careful delineations of the conformal transformations
required to ensure that a metric of Bianchi IX type has zero Ricci tensor, so
that it is a self-dual, vacuum solution of the complex-valued version of
Einstein's equations, as appropriate for the original Plebanski equation.Comment: 27 page
Partner symmetries and non-invariant solutions of four-dimensional heavenly equations
We extend our method of partner symmetries to the hyperbolic complex
Monge-Amp\`ere equation and the second heavenly equation of Pleba\~nski. We
show the existence of partner symmetries and derive the relations between them
for both equations. For certain simple choices of partner symmetries the
resulting differential constraints together with the original heavenly
equations are transformed to systems of linear equations by an appropriate
Legendre transformation. The solutions of these linear equations are
generically non-invariant. As a consequence we obtain explicitly new classes of
heavenly metrics without Killing vectors.Comment: 20 pages, 1 table, corrected typo
Partner symmetries of the complex Monge-Ampere equation yield hyper-Kahler metrics without continuous symmetries
We extend the Mason-Newman Lax pair for the elliptic complex Monge-Amp\`ere
equation so that this equation itself emerges as an algebraic consequence. We
regard the function in the extended Lax equations as a complex potential. We
identify the real and imaginary parts of the potential, which we call partner
symmetries, with the translational and dilatational symmetry characteristics
respectively. Then we choose the dilatational symmetry characteristic as the
new unknown replacing the K\"ahler potential which directly leads to a Legendre
transformation and to a set of linear equations satisfied by a single real
potential. This enables us to construct non-invariant solutions of the Legendre
transform of the complex Monge-Amp\`ere equation and obtain hyper-K\"ahler
metrics with anti-self-dual Riemann curvature 2-form that admit no Killing
vectors.Comment: submitted to J. Phys.
Lorentz and Galilei Invariance on Lattices
We show that the algebraic aspects of Lie symmetries and generalized
symmetries in nonrelativistic and relativistic quantum mechanics can be
preserved in linear lattice theories. The mathematical tool for symmetry
preserving discretizations on regular lattices is the umbral calculus.Comment: 5 page
Hamiltonian structure of real Monge-Amp\`ere equations
The real homogeneous Monge-Amp\`{e}re equation in one space and one time
dimensions admits infinitely many Hamiltonian operators and is completely
integrable by Magri's theorem. This remarkable property holds in arbitrary
number of dimensions as well, so that among all integrable nonlinear evolution
equations the real homogeneous Monge-Amp\`{e}re equation is distinguished as
one that retains its character as an integrable system in multi-dimensions.
This property can be traced back to the appearance of arbitrary functions in
the Lagrangian formulation of the real homogeneous Monge-Amp\`ere equation
which is degenerate and requires use of Dirac's theory of constraints for its
Hamiltonian formulation. As in the case of most completely integrable systems
the constraints are second class and Dirac brackets directly yield the
Hamiltonian operators. The simplest Hamiltonian operator results in the
Kac-Moody algebra of vector fields and functions on the unit circle.Comment: published in J. Phys. A 29 (1996) 325
Anticipated impact of the 2009 Four Corners raid and arrests
Archaeological looting on United States federal land has been illegal for over a century. Regardless, the activity has continued in the Four Corners region. This paper discusses how the 1979 Archaeological Resources Protection Act (ARPA) can be viewed as sumptuary law, and within a sumptuary context, subversion can be anticipated. An analysis of 1986 and June 2009 federal raids in the Four Corners will exemplify this point by identifying local discourses found in newspapers both before and after each raid, which demonstrate a sumptuary effect. Ultimately, this paper concludes that looting just adapted, rather than halted, after each federal raid and that understanding this social context of continued local justification and validation of illegal digging is a potential asset for cultural resource protection
Higher Order Quantum Superintegrability: a new "Painlev\'e conjecture"
We review recent results on superintegrable quantum systems in a
two-dimensional Euclidean space with the following properties. They are
integrable because they allow the separation of variables in Cartesian
coordinates and hence allow a specific integral of motion that is a second
order polynomial in the momenta. Moreover, they are superintegrable because
they allow an additional integral of order . Two types of such
superintegrable potentials exist. The first type consists of "standard
potentials" that satisfy linear differential equations. The second type
consists of "exotic potentials" that satisfy nonlinear equations. For , 4
and 5 these equations have the Painlev\'e property. We conjecture that this is
true for all . The two integrals X and Y commute with the Hamiltonian,
but not with each other. Together they generate a polynomial algebra (for any
) of integrals of motion. We show how this algebra can be used to calculate
the energy spectrum and the wave functions.Comment: 23 pages, submitted as a contribution to the monographic volume
"Integrability, Supersymmetry and Coherent States", a volume in honour of
Professor V\'eronique Hussin. arXiv admin note: text overlap with
arXiv:1703.0975
Unifying view of mechanical and functional hotspots across class A GPCRs
G protein-coupled receptors (GPCRs) are the largest superfamily of signaling proteins. Their activation process is accompanied by conformational changes that have not yet been fully uncovered. Here, we carry out a novel comparative analysis of internal structural fluctuations across a variety of receptors from class A GPCRs, which currently has the richest structural coverage. We infer the local mechanical couplings underpinning the receptors' functional dynamics and finally identify those amino acids whose virtual deletion causes a significant softening of the mechanical network. The relevance of these amino acids is demonstrated by their overlap with those known to be crucial for GPCR function, based on static structural criteria. The differences with the latter set allow us to identify those sites whose functional role is more clearly detected by considering dynamical and mechanical properties. Of these sites with a genuine mechanical/dynamical character, the top ranking is amino acid 7x52, a previously unexplored, and experimentally verifiable key site for GPCR conformational response to ligand binding. \ua9 2017 Ponzoni et al
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