5,405 research outputs found
Characterization of subdifferentials of a singular convex functional in Sobolev spaces of order minus one
Subdifferentials of a singular convex functional representing the surface
free energy of a crystal under the roughening temperature are characterized.
The energy functional is defined on Sobolev spaces of order -1, so the
subdifferential mathematically formulates the energy's gradient which formally
involves 4th order spacial derivatives of the surface's height. The
subdifferentials are analyzed in the negative Sobolev spaces of arbitrary
spacial dimension on which both a periodic boundary condition and a Dirichlet
boundary condition are separately imposed. Based on the characterization
theorem of subdifferentials, the smallest element contained in the
subdifferential of the energy for a spherically symmetric surface is calculated
under the Dirichlet boundary condition.Comment: 26 page
Noise Response Data Reveal Novel Controllability Gramian for Nonlinear Network Dynamics
Control of nonlinear large-scale dynamical networks, e.g., collective
behavior of agents interacting via a scale-free connection topology, is a
central problem in many scientific and engineering fields. For the linear
version of this problem, the so-called controllability Gramian has played an
important role to quantify how effectively the dynamical states are reachable
by a suitable driving input. In this paper, we first extend the notion of the
controllability Gramian to nonlinear dynamics in terms of the Gibbs
distribution. Next, we show that, when the networks are open to environmental
noise, the newly defined Gramian is equal to the covariance matrix associated
with randomly excited, but uncontrolled, dynamical state trajectories. This
fact theoretically justifies a simple Monte Carlo simulation that can extract
effectively controllable subdynamics in nonlinear complex networks. In
addition, the result provides a novel insight into the relationship between
controllability and statistical mechanics.Comment: 9 pages, 3 figures; to appear in Scientific Report
Resource Bounded Unprovability of Computational Lower Bounds
This paper introduces new notions of asymptotic proofs,
PT(polynomial-time)-extensions, PTM(polynomial-time Turing
machine)-omega-consistency, etc. on formal theories of arithmetic including PA
(Peano Arithmetic). This paper shows that P not= NP (more generally, any
super-polynomial-time lower bound in PSPACE) is unprovable in a
PTM-omega-consistent theory T, where T is a consistent PT-extension of PA. This
result gives a unified view to the existing two major negative results on
proving P not= NP, Natural Proofs and relativizable proofs, through the two
manners of characterization of PTM-omega-consistency. We also show that the
PTM-omega-consistency of T cannot be proven in any PTM-omega-consistent theory
S, where S is a consistent PT-extension of T.Comment: 78 page
Quantum Phase Transitions to Charge Order and Wigner Crystal Under Interplay of Lattice Commensurability and Long-Range Coulomb Interaction
Relationship among Wigner crystal, charge order and Mott insulator is studied
by the path-integral renormalization group method for two-dimensional lattices
with long-range Coulomb interaction. In contrast to Hartree-Fock results, the
solid stability drastically increases with lattice commensurability. The
transition to liquid occurs at the electron gas parameter for the
filling showing large reduction from in the continuum
limit. Correct account of quantum fluctuations are crucial to understand
charge-order stability generally observed only at simple fractional fillings
and nature of quantum liquids away from them.Comment: 4 pages including 7 figure
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