1,532 research outputs found
Cops and Robbers is EXPTIME-complete
We investigate the computational complexity of deciding whether k cops can
capture a robber on a graph G. In 1995, Goldstein and Reingold conjectured that
the problem is EXPTIME-complete when both G and k are part of the input; we
prove this conjecture.Comment: v2: updated figures and slightly clarified some minor point
Ehlers Transformations and String Effective Action
We explicitly obtain the generalization of the Ehlers transformation for
stationary axisymmetric Einstein equations to string theory. This is
accomplished by finding the twist potential corresponding to the moduli fields
in the effective two dimensional theory. Twist potential and symmetric moduli
are shown to transform under an which is a manifest symmetry of the
equations of motion. The non-trivial action of this is given by the
Ehlers transformation and belongs to the set .Comment: 13 pages, minor corrections, version to appear in Physics Letters
B35
Symmetries of Heterotic String Theory
We study the symmetries of the two dimensional Heterotic string theory by
following the approach of Kinnersley et al for the study of stationary-axially
symmetric Einstein-Maxwell equations. We identify the finite dimensional groups
and for the Einstein-Maxwell equations. We also give the
constructions for the infinite number of conserved currents and the affine
symmetry algebra in this formulation. The generalized Ehlers
and Harrison transformations are identified and a parallel between the infinite
dimensional symmetry algebra for the heterotic string case with that arise in the case of Einstein-Maxwell equations is pointed out.Comment: 26 pages, Few comments added, version to appear in Nuclear Physics
Integrals of Motion in the Two Killing Vector Reduction of General Relativity
We apply the inverse scattering method to the midi-superspace models that are
characterized by a two-parameter Abelian group of motions with two spacelike
Killing vectors. We present a formulation that simplifies the construction of
the soliton solutions of Belinski\v i and Zakharov. Furthermore, it enables us
to obtain the zero curvature formulation for these models. Using this, and
imposing periodic boundary conditions corresponding to the Gowdy models when
the spatial topology is a three torus , we show that the equation of
motion for the monodromy matrix is an evolution equation of the Heisenberg
type. Consequently, the eigenvalues of the monodromy matrix are the generating
functionals for the integrals of motion. Furthermore, we utilise a suitable
formulation of the transition matrix to obtain explicit expressions for the
integrals of motion. This involves recursion relations which arise in solving
an equation of Riccati type. In the case when the two Killing vectors are
hypersurface orthogonal the integrals of motion have a particularly simple
form.Comment: 20 pages, plain TeX, SU-GP-93/7-8, UM-P-93/7
Infinite-Dimensional Symmetries of Two-Dimensional Coset Models
It has long been appreciated that the toroidal reduction of any gravity or
supergravity to two dimensions gives rise to a scalar coset theory exhibiting
an infinite-dimensional global symmetry. This symmetry is an extension of the
finite-dimensional symmetry G in three dimensions, after performing a further
circle reduction. There has not been universal agreement as to exactly what the
extended symmetry algebra is, with different arguments seemingly concluding
either that it is , the affine Kac-Moody extension of G, or else a
subalgebra thereof. Exceptional in the literature for its explicit and
transparent exposition is the extremely lucid discussion by Schwarz, which we
take as our starting point for studying the simpler situation of
two-dimensional flat-space sigma models, which nonetheless capture all the
essential details. We arrive at the conclusion that the full symmetry is
described by the Kac-Moody algebra G, although truncations to subalgebras, such
as the one obtained by Schwarz, can be considered too. We then consider the
explicit example of the SL(2,R)/O(2) coset, and relate Schwarz's approach to an
earlier discussion that goes back to the work of Geroch.Comment: Typos corrected, some reorganisation; 36 page
Infinite-dimensional algebras in dimensionally reduced string theory
We examine 4-dimensional string backgrounds compactified over a two torus.
There exist two alternative effective Lagrangians containing each two
sigma-models. Two of these sigma-models are the complex and the
K\"ahler structures on the torus. The effective Lagrangians are invariant under
two different groups and by the successive applications of these
groups the affine Kac-Moody is emerged. The latter has also
a non-zero central term which generates constant Weyl rescalings of the reduced
2-dimensional background. In addition, there exists a number of discrete
symmetries relating the field content of the reduced effective Lagrangians.Comment: 13 pages, Late
Gravitational Solitons and Monodromy Transform Approach to Solution of Integrable Reductions of Einstein Equations
In this paper the well known Belinskii and Zakharov soliton generating
transformations of the solution space of vacuum Einstein equations with
two-dimensional Abelian groups of isometries are considered in the context of
the so called "monodromy transform approach", which provides some general base
for the study of various integrable space - time symmetry reductions of
Einstein equations. Similarly to the scattering data used in the known spectral
transform, in this approach the monodromy data for solution of associated
linear system characterize completely any solution of the reduced Einstein
equations, and many physical and geometrical properties of the solutions can be
expressed directly in terms of the analytical structure on the spectral plane
of the corresponding monodromy data functions. The Belinskii and Zakharov
vacuum soliton generating transformations can be expressed in explicit form
(without specification of the background solution) as simple
(linear-fractional) transformations of the corresponding monodromy data
functions with coefficients, polynomial in spectral parameter. This allows to
determine many physical parameters of the generating soliton solutions without
(or before) calculation of all components of the solutions. The similar
characterization for electrovacuum soliton generating transformations is also
presented.Comment: 8 pages, 1 figure, LaTeX2e; based on a talk given at the
International Conference 'Solitons, Collapses and Turbulence: Achievements,
Developments and Perspectives', (Landau Institute for Theoretical Physics,
Chernogolovka, Moscow region, Russia, August 3 -- 10, 1999); as submitted to
Physica
To catch a falling robber
We consider a Cops-and-Robber game played on the subsets of an -set. The
robber starts at the full set; the cops start at the empty set. On each turn,
the robber moves down one level by discarding an element, and each cop moves up
one level by gaining an element. The question is how many cops are needed to
ensure catching the robber when the robber reaches the middle level. Aaron Hill
posed the problem and provided a lower bound of for even and
for odd . We prove an
upper bound (for all ) that is within a factor of times this
lower bound.Comment: Minor revision
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