51,789 research outputs found
On Constructor Rewrite Systems and the Lambda Calculus
We prove that orthogonal constructor term rewrite systems and lambda-calculus
with weak (i.e., no reduction is allowed under the scope of a
lambda-abstraction) call-by-value reduction can simulate each other with a
linear overhead. In particular, weak call-by- value beta-reduction can be
simulated by an orthogonal constructor term rewrite system in the same number
of reduction steps. Conversely, each reduction in a term rewrite system can be
simulated by a constant number of beta-reduction steps. This is relevant to
implicit computational complexity, because the number of beta steps to normal
form is polynomially related to the actual cost (that is, as performed on a
Turing machine) of normalization, under weak call-by-value reduction.
Orthogonal constructor term rewrite systems and lambda-calculus are thus both
polynomially related to Turing machines, taking as notion of cost their natural
parameters.Comment: 27 pages. arXiv admin note: substantial text overlap with
arXiv:0904.412
Tangent-space methods for uniform matrix product states
In these lecture notes we give a technical overview of tangent-space methods for matrix product states in the thermodynamic limit. We introduce the manifold of uniform matrix product states, show how to compute different types of observables, and discuss the concept of a tangent space. We explain how to variationally optimize ground-state approximations, implement real-time evolution and describe elementary excitations for a given model Hamiltonian. Also, we explain how matrix product states approximate fixed points of one-dimensional transfer matrices. We show how all these methods can be translated to the language of continuous matrix product states for one-dimensional field theories. We conclude with some extensions of the tangent-space formalism and with an outlook to new applications
Degenerate Variational Integrators for Magnetic Field Line Flow and Guiding Center Trajectories
Symplectic integrators offer many advantages for the numerical solution of
Hamiltonian differential equations, including bounded energy error and the
preservation of invariant sets. Two of the central Hamiltonian systems
encountered in plasma physics --- the flow of magnetic field lines and the
guiding center motion of magnetized charged particles --- resist symplectic
integration by conventional means because the dynamics are most naturally
formulated in non-canonical coordinates, i.e., coordinates lacking the familiar
partitioning. Recent efforts made progress toward non-canonical
symplectic integration of these systems by appealing to the variational
integration framework; however, those integrators were multistep methods and
later found to be numerically unstable due to parasitic mode instabilities.
This work eliminates the multistep character and, therefore, the parasitic mode
instabilities via an adaptation of the variational integration formalism that
we deem ``degenerate variational integration''. Both the magnetic field line
and guiding center Lagrangians are degenerate in the sense that their resultant
Euler-Lagrange equations are systems of first-order ODEs. We show that
retaining the same degree of degeneracy when constructing a discrete Lagrangian
yields one-step variational integrators preserving a non-canonical symplectic
structure on the original Hamiltonian phase space. The advantages of the new
algorithms are demonstrated via numerical examples, demonstrating superior
stability compared to existing variational integrators for these systems and
superior qualitative behavior compared to non-conservative algorithms
Extending Context-Sensitivity in Term Rewriting
We propose a generalized version of context-sensitivity in term rewriting
based on the notion of "forbidden patterns". The basic idea is that a rewrite
step should be forbidden if the redex to be contracted has a certain shape and
appears in a certain context. This shape and context is expressed through
forbidden patterns. In particular we analyze the relationships among this novel
approach and the commonly used notion of context-sensitivity in term rewriting,
as well as the feasibility of rewriting with forbidden patterns from a
computational point of view. The latter feasibility is characterized by
demanding that restricting a rewrite relation yields an improved termination
behaviour while still being powerful enough to compute meaningful results.
Sufficient criteria for both kinds of properties in certain classes of rewrite
systems with forbidden patterns are presented
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