Nongeneric null vectors

We consider the generic condition for null directions at a fixed point. A nongeneric vector is one which violates the generic condition: a vectorX is nongeneric ifX ^cX^d X[_aR_b]cd[_e X _f] = 0. The presence of null nongeneric vectors at a point can force the curvature tensor to be uniform, i.e., be that of a constant-curvature space; in particular, if there are 11 null nongeneric directions generically situated, then the curvature is uniform. For non-uniform curvature, the locus of nongeneric null directions must obey a cubic relation; examples show that it need not obey a linear relation.     

Universe admitting symmetric vectors

The possible number and location of symmetric vectors in curved space is determined. For the case of isotropic universe (of both positive and negative constant curvature) all the symmetric vectors are explicitly constructed.     

Checking collineation vectors with reduce

A package of programs for writing and checking the solutions to the equations for various types of collineations, written in the symbolic manipulation language REDUCE is presented. Some examples of previously found collineations that have been checked are given, and new results reported.     

Quantum contextuality for rational vectors

The Kochen-Specker theorem states that noncontextual hidden variable models are inconsistent with the quantum predictions for every yes-no question on a qutrit, corresponding to every projector in three dimensions. It has been suggested [D.A. Meyer, Phys. Rev. Lett. 83 (1999) 3751] that the inconsistency would disappear when restricting to projectors on unit vectors with rational components; that noncontextual hidden variables could reproduce the quantum predictions for rational vectors. Here we show that a qutrit state with rational components violates an inequality valid for noncontextual hidden-variable models [A.A. Klyachko et al., Phys. Rev. Lett. 101 (2008) 020403] using rational projectors. This shows that the inconsistency remains even when using only rational vectors.     

Second-order systems with Runge-Lenz-type vectors

A class of second-order equation fields which contains the equations of motion of the Kepler and charge monopole problems is considered. In particular, the problems of the existence of a second integral of motion quadratic in momenta besides the energy is investigated and completely solved. A subclass of systems is exhibited which have a Runge-Lenz-type vector, but it is shown that the monopole system itself has no second (time-independent) quadratic integral of motion.     

Optical helicity and Hertz vectors

We study the conserved quantity associated with the dual symmetry of the Maxwell equations, called the optical helicity, by means of transverse Hertz vectors. In the presence of charges, its evolution yields the integral of $\mathit{E}?\mathit{B}$ which is the anomaly term for chiral fermions. We also discuss the helicity change in condensed matter systems where topological magnetoelectric effect emerges. An alternative expression of the optical helicity is also found. Lastly, a dual symmetric Hertz Lagrangian is constructed and its conserved charge is derived.     

Conserved quantities from piecewise Killing vectors

In the presence of symmetries, conserved quantities can be obtained by contracting the stress-energy tensor with a Killing vector. We generalize this result to piecewise Killing vectors by giving sufficient conditions for the construction of an associated conserved quantity. A typical example, namely, two stationary space-times joined together in such a way that the resulting space-time is not stationary, is treated in detail.     

Noise operators and approximate state vectors

A recent paper of Dekker on the noise operator approach to quantized dissipative systems is examined in some detail. It is shown that the occurrence of the noise operators is intimately related to a possible approximate Schrödinger picture representation for the state vector of the macroscopic quantum system. To do this, we have assumed that an initially pure collective state, which in our case is the macroscopic superfluid ground state, remains a pure state under a given condition. Then, we develop the theory by introducing a special type of time derivatives, which are akin to the covariant derivatives in the theory of relativity.     

Szekeres's space-times have no Killing vectors

Szekeres obtained some exact solutions of Einstein's equations for dust which have applications in cosmology, in the theory of collapse, and in the study of gravitational radiation. It is shown that in their most general form these space-times have no Killing vectors.Dedicated to Achille Papapetrou on the occasion of his retirement.On leave from the Department of Mathematics, University of Malaya, Kuala Lumpur, Malaysia.Supported by the Public Services Department, Government of Malaysia.On leave from Centro Brasileiro de Pesquisas Fisicas.     

Singular vectors of the Virasoro algebra

We present an explicit construction of the singular (or null) vectors in highest weight Verma modules.     

On the structure of Bethe vectors

The structure of Bethe vectors for generalised models associated with the rational and trigonometric R-matrix is investigated. The Bethe vectors in terms of two-component and multi-component models are described. Consequently, their structure in terms of local variables and operators is provided. This, as a consequence, proves the equivalence of coordinate and algebraic Bethe ansatzes for the Heisenberg spin chains. Hermitian conjugation of the elements of the monodromy matrix for the spin chains is studied.     

Quasi invariants and generalized killing vectors

The connection between quasi-invariants (invariants of a Hamiltonian system defined only on a single constant energy hypersurface) and generalized Killing vector fields associated with the corresponding Jacobi metric is investigated. The results are used to deduce a generalised form of the classical Whittaker problem in two degrees of freedom.     

Invariant conformal vectors in static spacetimes

Static spacetimes are studied in terms of a space-plus-time decomposition proposed by Petrov. The problem of finding invariant conformal vectors (the ones which commute with the static Killing vector) is reduced to the simpler problem of finding the isometries of a three-dimensional metric. In the perfect fluid case, all the line elements admitting proper invariant conformal vectors are explicitly given.     

Conformal Killing vectors and FRW spacetimes

Fluid space-times which admit a conformal Killing vector (CKV) are studied. It is shown that even in a perfect fluid space-time a conformal motion will not, in general, map the fluid flow lines onto fluid flow lines; consequently, perfect fluid space-times and, in particular, the simplest perfect fluid space-times known to admit a CKV, namely the Friedmann-Robertson-Walker (FRW) space-times, are studied. A direct proof that there do not exist any special CKV in FRW space-times will be given, thereby motivating the study of the physically more relevant proper CKV. Indeed, one of the principal motivations of the present work is the study of the symmetry inheritance problem for proper CKV. Since the FRW metric can, in general, satisfy the Einstein field equations for a non-comoving imperfect fluid, the relationship between the FRW models (and in particular the standard comoving perfect fluid models) and the conditions under which conformal motions (and in addition homothetic motions) map fluid flow lines onto fluid flow lines are investigated. Finally, further properties of fluid space-times which admit a proper CKV, and in particular space-times in which the CKV is parallel to the fluid four-velocity, are discussed.     

Characterizing dynamics with covariant Lyapunov vectors

A general method to determine covariant Lyapunov vectors in both discrete- and continuous-time dynamical systems is introduced. This allows us to address fundamental questions such as the degree of hyperbolicity, which can be quantified in terms of the transversality of these intrinsic vectors. For spatially extended systems, the covariant Lyapunov vectors have localization properties and spatial Fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the Lyapunov exponents.     

Partial transfer entropy on rank vectors

For the evaluation of information flow in bivariate time series, information measures have been employed, such as the transfer entropy (TE), the symbolic transfer entropy (STE), defined similarly to TE but on the ranks of the components of the reconstructed vectors, and the transfer entropy on rank vectors (TERV), similar to STE but forming the ranks for the future samples of the response system with regard to the current reconstructed vector. Here we extend TERV for multivariate time series, and account for the presence of confounding variables, called partial transfer entropy on ranks (PTERV). We investigate the asymptotic properties of PTERV, and also partial STE (PSTE), construct parametric significance tests under approximations with Gaussian and gamma null distributions, and show that the parametric tests cannot achieve the power of the randomization test using time-shifted surrogates. Using simulations on known coupled dynamical systems and applying parametric and randomization significance tests, we show that PTERV performs better than PSTE but worse than the partial transfer entropy (PTE). However, PTERV, unlike PTE, is robust to the presence of drifts in the time series and it is also not affected by the level of detrending.     

Noise operators and approximate state vectors

A recent paper of Dekker on the noise operator approach to quantized dissipative systems is examined in some detail. It is shown that the occurrence of the noise operators is intimately related to a possible approximate Schrödinger picture representation for the state vector of the macroscopic quantum system. To do this, we have assumed that an initially pure collective state, which in our case is the macroscopic superfluid ground state, remains a pure state under a given condition. Then, we develop the theory by introducing a special type of time derivatives, which are akin to the covariant derivatives in the theory of relativity.     

General construction for topological singular vectors

A general construction is found for “topological” singular vectors (BRST-invariant singular vectors of the twisted N=2 superconformal algebra which are constructed over chiral primary states). The new construction, employed in parallel with the well-known MFF construction for the singular vectors of the sℓ(2) Kac-Moody algebra, implements an isomorphism of topological and sℓ(2)-singular vectors. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 2, 129–134 (25 January 1996)     

Singular vectors of the Neveu-Schwarz algebra

We give an explicit expression for some singular vectors of highest weight representations of the Neveu-Schwarz algebra.