2,278 research outputs found
Solution of polynomial Lyapunov and Sylvester equations
A two-variable polynomial approach to solve the one-variable polynomial Lyapunov and Sylvester equations is proposed. Lifting the problem from the one-variable to the two-variable context gives rise to associated lifted equations which live on finite-dimensional vector spaces. This allows for the design of an iterative solution method which is inspired by the method of Faddeev for the computation of matrix resolvents. The resulting algorithms are especially suitable for applications requiring symbolic or exact computation
B\"acklund Transformation for the BC-Type Toda Lattice
We study an integrable case of n-particle Toda lattice: open chain with
boundary terms containing 4 parameters. For this model we construct a
B\"acklund transformation and prove its basic properties: canonicity,
commutativity and spectrality. The B\"acklund transformation can be also viewed
as a discretized time dynamics. Two Lax matrices are used: of order 2 and of
order 2n+2, which are mutually dual, sharing the same spectral curve.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Enumeration of integral tetrahedra
We determine the numbers of integral tetrahedra with diameter up to
isomorphism for all via computer enumeration. Therefore we give an
algorithm that enumerates the integral tetrahedra with diameter at most in
time and an algorithm that can check the canonicity of a given
integral tetrahedron with at most 6 integer comparisons. For the number of
isomorphism classes of integral matrices with diameter
fulfilling the triangle inequalities we derive an exact formula.Comment: 10 pages, 1 figur
Solution of the noncanonicity puzzle in General Relativity: a new Hamiltonian formulation
We study the transformation leading from Arnowitt, Deser, Misner (ADM)
Hamiltonian formulation of General Relativity (GR) to the metric
Hamiltonian formulation derived from the Lagrangian density which was firstly
proposed by Einstein. We classify this transformation as gauged canonical -
i.e. canonical modulo a gauge transformation. In such a study we introduce a
new Hamiltonian formulation written in ADM variables which differs from the
usual ADM formulation mainly in a boundary term firstly proposed by Dirac.
Performing the canonical quantization procedure we introduce a new functional
phase which contains an explicit dependence on the fields characterizing the
3+1 splitting. Given a specific regularization procedure our new formulation
privileges the symmetric operator ordering in order to: have a consistent
quantization procedure, avoid anomalies in constraints algebra, be equivalent
to the Wheeler-DeWitt (WDW) quantization. Furthermore we show that this result
is consistent with a path-integral approach.Comment: Accepted for Publication in Physics Letters B. Major revisions in
Canonical Quantization section for operator ordering choice and in the
definition of 'gauged canonicity' for the classical analysis. 19 pages single
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