157,697 research outputs found
Unfreezing Casimir invariants: singular perturbations giving rise to forbidden instabilities
The infinite-dimensional mechanics of fluids and plasmas can be formulated as
"noncanonical" Hamiltonian systems on a phase space of Eulerian variables.
Singularities of the Poisson bracket operator produce singular Casimir elements
that foliate the phase space, imposing topological constraints on the dynamics.
Here we proffer a physical interpretation of Casimir elements as
\emph{adiabatic invariants} ---upon coarse graining microscopic angle
variables, we obtain a macroscopic hierarchy on which the separated action
variables become adiabatic invariants. On reflection, a Casimir element may be
\emph{unfrozen} by recovering a corresponding angle variable; such an increase
in the number of degrees of freedom is, then, formulated as a \emph{singular
perturbation}. As an example, we propose a canonization of the
resonant-singularity of the Poisson bracket operator of the linearized
magnetohydrodynamics equations, by which the ideal obstacle (resonant Casimir
element) constraining the dynamics is unfrozen, giving rise to a tearing-mode
instability
General Rule and Materials Design of Negative Effective U System for High-T_c Superconductivity
Based on the microscopic mechanisms of (1) charge-excitation-induced negative
effective U in s^1 or d^9 electronic configurations, and (2)
exchange-correlation-induced negative effective U in d^4 or d^6 electronic
configurations, we propose a general rule and materials design of negative
effective U system in itinerant (ionic and metallic) system for the realization
of high-T_c superconductors. We design a T_c-enhancing layer (or clusters) of
charge-excitation-induced negative effective connecting the superconducting
layers for the realistic systems.Comment: 11 pages, 1 figures, 2 tables, APEX in printin
Optimal Constant-Time Approximation Algorithms and (Unconditional) Inapproximability Results for Every Bounded-Degree CSP
Raghavendra (STOC 2008) gave an elegant and surprising result: if Khot's
Unique Games Conjecture (STOC 2002) is true, then for every constraint
satisfaction problem (CSP), the best approximation ratio is attained by a
certain simple semidefinite programming and a rounding scheme for it. In this
paper, we show that similar results hold for constant-time approximation
algorithms in the bounded-degree model. Specifically, we present the
followings: (i) For every CSP, we construct an oracle that serves an access, in
constant time, to a nearly optimal solution to a basic LP relaxation of the
CSP. (ii) Using the oracle, we give a constant-time rounding scheme that
achieves an approximation ratio coincident with the integrality gap of the
basic LP. (iii) Finally, we give a generic conversion from integrality gaps of
basic LPs to hardness results. All of those results are \textit{unconditional}.
Therefore, for every bounded-degree CSP, we give the best constant-time
approximation algorithm among all. A CSP instance is called -far from
satisfiability if we must remove at least an -fraction of constraints
to make it satisfiable. A CSP is called testable if there is a constant-time
algorithm that distinguishes satisfiable instances from -far
instances with probability at least . Using the results above, we also
derive, under a technical assumption, an equivalent condition under which a CSP
is testable in the bounded-degree model
Testing List H-Homomorphisms
Let be an undirected graph. In the List -Homomorphism Problem, given
an undirected graph with a list constraint for each
variable , the objective is to find a list -homomorphism , that is, for every and whenever .
We consider the following problem: given a map as an oracle
access, the objective is to decide with high probability whether is a list
-homomorphism or \textit{far} from any list -homomorphisms. The
efficiency of an algorithm is measured by the number of accesses to .
In this paper, we classify graphs with respect to the query complexity
for testing list -homomorphisms and show the following trichotomy holds: (i)
List -homomorphisms are testable with a constant number of queries if and
only if is a reflexive complete graph or an irreflexive complete bipartite
graph. (ii) List -homomorphisms are testable with a sublinear number of
queries if and only if is a bi-arc graph. (iii) Testing list
-homomorphisms requires a linear number of queries if is not a bi-arc
graph
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