10,223 research outputs found
Affine Extensions of Integer Vector Addition Systems with States
We study the reachability problem for affine -VASS, which are
integer vector addition systems with states in which transitions perform affine
transformations on the counters. This problem is easily seen to be undecidable
in general, and we therefore restrict ourselves to affine -VASS
with the finite-monoid property (afmp--VASS). The latter have the
property that the monoid generated by the matrices appearing in their affine
transformations is finite. The class of afmp--VASS encompasses
classical operations of counter machines such as resets, permutations,
transfers and copies. We show that reachability in an afmp--VASS
reduces to reachability in a -VASS whose control-states grow
linearly in the size of the matrix monoid. Our construction shows that
reachability relations of afmp--VASS are semilinear, and in
particular enables us to show that reachability in -VASS with
transfers and -VASS with copies is PSPACE-complete. We then focus
on the reachability problem for affine -VASS with monogenic
monoids: (possibly infinite) matrix monoids generated by a single matrix. We
show that, in a particular case, the reachability problem is decidable for this
class, disproving a conjecture about affine -VASS with infinite
matrix monoids we raised in a preliminary version of this paper. We complement
this result by presenting an affine -VASS with monogenic matrix
monoid and undecidable reachability relation
Galois Unitaries, Mutually Unbiased Bases, and MUB-balanced states
A Galois unitary is a generalization of the notion of anti-unitary operators.
They act only on those vectors in Hilbert space whose entries belong to some
chosen number field. For Mutually Unbiased Bases the relevant number field is a
cyclotomic field. By including Galois unitaries we are able to remove a
mismatch between the finite projective group acting on the bases on the one
hand, and the set of those permutations of the bases that can be implemented as
transformations in Hilbert space on the other hand. In particular we show that
there exist transformations that cycle through all the bases in every dimension
which is an odd power of an odd prime. (For even primes unitary MUB-cyclers
exist.) These transformations have eigenvectors, which are MUB-balanced states
(i.e. rotationally symmetric states in the original terminology of Wootters and
Sussman) if and only if d = 3 modulo 4. We conjecture that this construction
yields all such states in odd prime power dimension.Comment: 32 pages, 2 figures, AMS Latex. Version 2: minor improvements plus a
few additional reference
Superstring Theory on pp Waves with ADE Geometries
We study the BMN correspondence between certain Penrose limits of type IIB
superstrings on pp-wave orbifolds with geometries, and the set of
four-dimensional superconformal field theories constructed as
quiver gauge models classified by finite Lie algebras and affine
Kac-Moody algebras. These models have 16 preserved supercharges and
are based on systems of D3-branes and wrapped D5- and D7-branes. We derive
explicitly the metrics of these pp-wave orbifolds and show that the BMN
extension requires, in addition to D5-D5 open strings in bi-fundamental
representations, D5-D7 open strings involving orientifolds with gauge
symmetry. We also give the correspondence rule between leading string states
and gauge-invariant operators in the quiver gauge models.Comment: 62 page
Standard and Non-standard Extensions of Lie algebras
We study the problem of quadruple extensions of simple Lie algebras. We find
that, adding a new simple root , it is not possible to have an
extended Kac-Moody algebra described by a Dynkin-Kac diagram with simple links
and no loops between the dots, while it is possible if is a
Borcherds imaginary simple root. We also comment on the root lattices of these
new algebras. The folding procedure is applied to the simply-laced triple
extended Lie algebras, obtaining all the non-simply laced ones. Non- standard
extension procedures for a class of Lie algebras are proposed. It is shown that
the 2-extensions of , with a dot simply linked to the Dynkin-Kac diagram
of , are rank 10 subalgebras of . Finally the simple root
systems of a set of rank 11 subalgebras of , containing as sub-algebra
, are explicitly written.Comment: Revised version. Inaccurate statements corrected. Expanded version
with added reference
Integer Vector Addition Systems with States
This paper studies reachability, coverability and inclusion problems for
Integer Vector Addition Systems with States (ZVASS) and extensions and
restrictions thereof. A ZVASS comprises a finite-state controller with a finite
number of counters ranging over the integers. Although it is folklore that
reachability in ZVASS is NP-complete, it turns out that despite their
naturalness, from a complexity point of view this class has received little
attention in the literature. We fill this gap by providing an in-depth analysis
of the computational complexity of the aforementioned decision problems. Most
interestingly, it turns out that while the addition of reset operations to
ordinary VASS leads to undecidability and Ackermann-hardness of reachability
and coverability, respectively, they can be added to ZVASS while retaining
NP-completness of both coverability and reachability.Comment: 17 pages, 2 figure
Subnormal operators regarded as generalized observables and compound-system-type normal extension related to su(1,1)
In this paper, subnormal operators, not necessarily bounded, are discussed as
generalized observables. In order to describe not only the information about
the probability distribution of the output data of their measurement but also a
framework of their implementations, we introduce a new concept
compound-system-type normal extension, and we derive the compound-system-type
normal extension of a subnormal operator, which is defined from an irreducible
unitary representation of the algebra su(1,1). The squeezed states are
characterized as the eigenvectors of an operator from this viewpoint, and the
squeezed states in multi-particle systems are shown to be the eigenvectors of
the adjoints of these subnormal operators under a representation. The affine
coherent states are discussed in the same context, as well.Comment: LaTeX with iopart.cls, iopart12.clo, iopams.sty, The previous version
has some mistake
Composable computation in discrete chemical reaction networks
We study the composability of discrete chemical reaction networks (CRNs) that
stably compute (i.e., with probability 0 of error) integer-valued functions
. We consider output-oblivious CRNs in which the
output species is never a reactant (input) to any reaction. The class of
output-oblivious CRNs is fundamental, appearing in earlier studies of CRN
computation, because it is precisely the class of CRNs that can be composed by
simply renaming the output of the upstream CRN to match the input of the
downstream CRN.
Our main theorem precisely characterizes the functions stably computable
by output-oblivious CRNs with an initial leader. The key necessary condition is
that for sufficiently large inputs, is the minimum of a finite number of
nondecreasing quilt-affine functions. (An affine function is linear with a
constant offset; a quilt-affine function is linear with a periodic offset)
- …