265,143 research outputs found
Periodic subvarieties of a projective variety under the action of a maximal rank abelian group of positive entropy
We determine positive-dimensional G-periodic proper subvarieties of an
n-dimensional normal projective variety X under the action of an abelian group
G of maximal rank n-1 and of positive entropy. The motivation of the paper is
to understand the obstruction for X to be G-equivariant birational to the
quotient variety of an abelian variety modulo the action of a finite group.Comment: Asian Journal of Mathematics (to appear), Special issue on the
occasion of Prof N. Mok's 60th birthda
Inverse Statistical Mechanics: Probing the Limitations of Isotropic Pair Potentials to Produce Ground-State Structural Extremes
Inverse statistical-mechanical methods have recently been employed to design
optimized short-ranged radial (isotropic) pair potentials that robustly produce
novel targeted classical ground-state many-particle configurations. The target
structures considered in those studies were low-coordinated crystals with a
high degree of symmetry. In this paper, we further test the fundamental
limitations of radial pair potentials by targeting crystal structures with
appreciably less symmetry, including those in which the particles have
different local structural environments. These challenging target
configurations demanded that we modify previous inverse optimization
techniques. Using this modified optimization technique, we have designed
short-ranged radial pair potentials that stabilize the two-dimensional kagome
crystal, the rectangular kagome crystal, and rectangular lattices, as well as
the three-dimensional structure of CaF crystal inhabited by a single
particle species. We verify our results by cooling liquid configurations to
absolute zero temperature via simulated annealing and ensuring that such states
have stable phonon spectra. Except for the rectangular kagome structure, all of
the target structures can be stabilized with monotonic repulsive potentials.
Our work demonstrates that single-component systems with short-ranged radial
pair potentials can counterintuitively self-assemble into crystal ground states
with low symmetry and different local structural environments. Finally, we
present general principles that offer guidance in determining whether certain
target structures can be achieved as ground states by radial pair potentials
Classical many-particle systems with unique disordered ground states
Classical ground states (global energy-minimizing configurations) of
many-particle systems are typically unique crystalline structures, implying
zero enumeration entropy of distinct patterns (aside from trivial symmetry
operations). By contrast, the few previously known disordered classical ground
states of many-particle systems are all high-entropy (highly degenerate)
states. Here we show computationally that our recently-proposed "perfect-glass"
many-particle model [Sci. Rep., 6, 36963 (2016)] possesses disordered classical
ground states with a zero entropy: a highly counterintuitive situation. For all
of the system sizes, parameters, and space dimensions that we have numerically
investigated, the disordered ground states are unique such that they can always
be superposed onto each other or their mirror image. At low energies, the
density of states obtained from simulations matches those calculated from the
harmonic approximation near a single ground state, further confirming
ground-state uniqueness. Our discovery provides singular examples in which
entropy and disorder are at odds with one another. The zero-entropy ground
states provide a unique perspective on the celebrated Kauzmann-entropy crisis
in which the extrapolated entropy of a supercooled liquid drops below that of
the crystal. We expect that our disordered unique patterns to be of value in
fields beyond glass physics, including applications in cryptography as
pseudo-random functions with tunable computational complexity
Transport, Geometrical and Topological Properties of Stealthy Disordered Hyperuniform Two-Phase Systems
Disordered hyperuniform many-particle systems have attracted considerable
recent attention. One important class of such systems is the classical ground
states of "stealthy potentials." The degree of order of such ground states
depends on a tuning parameter. Previous studies have shown that these
ground-state point configurations can be counterintuitively disordered,
infinitely degenerate, and endowed with novel physical properties (e.g.,
negative thermal expansion behavior). In this paper, we focus on the disordered
regime in which there is no long-range order, and control the degree of
short-range order. We map these stealthy disordered hyperuniform point
configurations to two-phase media by circumscribing each point with a possibly
overlapping sphere of a common radius : the "particle" and "void" phases are
taken to be the space interior and exterior to the spheres, respectively. We
study certain transport properties of these systems, including the effective
diffusion coefficient of point particles diffusing in the void phase as well as
static and time-dependent characteristics associated with diffusion-controlled
reactions. Besides these effective transport properties, we also investigate
several related structural properties, including pore-size functions, quantizer
error, an order metric, and percolation threshold. We show that these
transport, geometrical and topological properties of our two-phase media
derived from decorated stealthy ground states are distinctly different from
those of equilibrium hard-sphere systems and spatially uncorrelated overlapping
spheres
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