2,949 research outputs found
Rendezvous of Two Robots with Constant Memory
We study the impact that persistent memory has on the classical rendezvous
problem of two mobile computational entities, called robots, in the plane. It
is well known that, without additional assumptions, rendezvous is impossible if
the entities are oblivious (i.e., have no persistent memory) even if the system
is semi-synchronous (SSynch). It has been recently shown that rendezvous is
possible even if the system is asynchronous (ASynch) if each robot is endowed
with O(1) bits of persistent memory, can transmit O(1) bits in each cycle, and
can remember (i.e., can persistently store) the last received transmission.
This setting is overly powerful.
In this paper we weaken that setting in two different ways: (1) by
maintaining the O(1) bits of persistent memory but removing the communication
capabilities; and (2) by maintaining the O(1) transmission capability and the
ability to remember the last received transmission, but removing the ability of
an agent to remember its previous activities. We call the former setting
finite-state (FState) and the latter finite-communication (FComm). Note that,
even though its use is very different, in both settings, the amount of
persistent memory of a robot is constant.
We investigate the rendezvous problem in these two weaker settings. We model
both settings as a system of robots endowed with visible lights: in FState, a
robot can only see its own light, while in FComm a robot can only see the other
robot's light. We prove, among other things, that finite-state robots can
rendezvous in SSynch, and that finite-communication robots are able to
rendezvous even in ASynch. All proofs are constructive: in each setting, we
present a protocol that allows the two robots to rendezvous in finite time.Comment: 18 pages, 3 figure
The Random Bit Complexity of Mobile Robots Scattering
We consider the problem of scattering robots in a two dimensional
continuous space. As this problem is impossible to solve in a deterministic
manner, all solutions must be probabilistic. We investigate the amount of
randomness (that is, the number of random bits used by the robots) that is
required to achieve scattering. We first prove that random bits are
necessary to scatter robots in any setting. Also, we give a sufficient
condition for a scattering algorithm to be random bit optimal. As it turns out
that previous solutions for scattering satisfy our condition, they are hence
proved random bit optimal for the scattering problem. Then, we investigate the
time complexity of scattering when strong multiplicity detection is not
available. We prove that such algorithms cannot converge in constant time in
the general case and in rounds for random bits optimal
scattering algorithms. However, we present a family of scattering algorithms
that converge as fast as needed without using multiplicity detection. Also, we
put forward a specific protocol of this family that is random bit optimal ( random bits are used) and time optimal ( rounds are used).
This improves the time complexity of previous results in the same setting by a
factor. Aside from characterizing the random bit complexity of mobile
robot scattering, our study also closes its time complexity gap with and
without strong multiplicity detection (that is, time complexity is only
achievable when strong multiplicity detection is available, and it is possible
to approach it as needed otherwise)
Moebius Structure of the Spectral Space of Schroedinger Operators with Point Interaction
The Schroedinger operator with point interaction in one dimension has a U(2)
family of self-adjoint extensions. We study the spectrum of the operator and
show that (i) the spectrum is uniquely determined by the eigenvalues of the
matrix U belonging to U(2) that characterizes the extension, and that (ii) the
space of distinct spectra is given by the orbifold T^2/Z_2 which is a Moebius
strip with boundary. We employ a parametrization of U(2) that admits a direct
physical interpretation and furnishes a coherent framework to realize the
spectral duality and anholonomy recently found. This allows us to find that
(iii) physically distinct point interactions form a three-parameter quotient
space of the U(2) family.Comment: 16 pages, 2 figure
Mathematical analysis of a model of river channel formation.
The study of overland flow of water over an erodible sediment leads to a coupled model describing the evolution of the topographic elevation and the depth of the overland water film. The spatially uniform solution of this model is unstable, and this instability corresponds to the formation of rills, which in reality then grow and coalesce to form large-scale river channels. In this paper we consider the deduction and mathematical analysis of a deterministic model describing river channel formation and the evolution of its depth. The model involves a degenerate nonlinear parabolic equation (satisfied on the interior of the support of the solution) with a super-linear source term and a prescribed constant mass. We propose here a global formulation of the problem (formulated in the whole space, beyond the support of the solution) which allows us to show the existence of a solution and leads to a suitable numerical scheme for its approximation. A particular novelty of the model is that the evolving channel self-determines its own width, without the need to pose any extra conditions at the channel margin
Spherically symmetric analysis on open FLRW solution in non-linear massive gravity
We study non-linear massive gravity in the spherically symmetric context. Our
main motivation is to investigate the effect of helicity-0 mode which remains
elusive after analysis of cosmological perturbation around an open
Friedmann-Lemaitre-Robertson-Walker (FLRW) universe. The non-linear form of the
effective energy-momentum tensor stemming from the mass term is derived for the
spherically symmetric case. Only in the special case where the area of the two
sphere is not deviated away from the FLRW universe, the effective energy
momentum tensor becomes completely the same as that of cosmological constant.
This opens a window for discriminating the non-linear massive gravity from
general relativity (GR). Indeed, by further solving these spherically symmetric
gravitational equations of motion in vacuum to the linear order, we obtain a
solution which has an arbitrary time-dependent parameter. In GR, this parameter
is a constant and corresponds to the mass of a star. Our result means that
Birkhoff's theorem no longer holds in the non-linear massive gravity and
suggests that energy can probably be emitted superluminously (with infinite
speed) on the self-accelerating background by the helicity-0 mode, which could
be a potential plague of this theory.Comment: 16 pages, references added; minor modification to match version
published in JCA
Exploration of finite dimensional Kac algebras and lattices of intermediate subfactors of irreducible inclusions
We study the four infinite families KA(n), KB(n), KD(n), KQ(n) of finite
dimensional Hopf (in fact Kac) algebras constructed respectively by A. Masuoka
and L. Vainerman: isomorphisms, automorphism groups, self-duality, lattices of
coideal subalgebras. We reduce the study to KD(n) by proving that the others
are isomorphic to KD(n), its dual, or an index 2 subalgebra of KD(2n). We
derive many examples of lattices of intermediate subfactors of the inclusions
of depth 2 associated to those Kac algebras, as well as the corresponding
principal graphs, which is the original motivation.
Along the way, we extend some general results on the Galois correspondence
for depth 2 inclusions, and develop some tools and algorithms for the study of
twisted group algebras and their lattices of coideal subalgebras. This research
was driven by heavy computer exploration, whose tools and methodology we
further describe.Comment: v1: 84 pages, 13 figures, submitted. v2: 94 pages, 15 figures, added
connections with Masuoka's families KA and KB, description of K3 in KD(n),
lattices for KD(8) and KD(15). v3: 93 pages, 15 figures, proven lattice for
KD(6), misc improvements, accepted for publication in Journal of Algebra and
Its Application
Helium Nova on a Very Massive White Dwarf -- A Light Curve Model of V445 Puppis (2000) Revised
V445 Pup (2000) is a unique object identified as a helium nova. Color indexes
during the outburst are consistent with those of free-free emission. We present
a free-free emission dominated light curve model of V445 Pup on the basis of
the optically thick wind theory. Our light curve fitting shows that (1) the
white dwarf (WD) mass is very massive (M_WD \gtrsim 1.35 M_\sun), and (2) a
half of the accreted matter remains on the WD, both of which suggest that the
increasing WD mass. Therefore, V445 Pup is a strong candidate of Type Ia
supernova progenitor. The estimated distance to V445 Pup is now consistent with
the recent observational suggestions, 3.5 < d < 6.5 kpc. A helium star
companion is consistent with the brightness of m_v=14.5 mag just before the
outburst, if it is a little bit evolved hot (\log T (K) \gtrsim 4.5) star with
the mass of M_He \gtrsim 0.8 M_\sun. We then emphasize importance of
observations in the near future quiescent phase after the thick circumstellar
dust dissipates away, especially its color and magnitude to specify the nature
of the companion star. We have also calculated helium ignition masses for
helium shell flashes against various helium accretion rates and discussed the
recurrence period of helium novae.Comment: 8 pages including 12 figures, to appear in Ap
Entanglement Measures for Intermediate Separability of Quantum States
We present a family of entanglement measures R_m which act as indicators for
separability of n-qubit quantum states into m subsystems for arbitrary 2 \leq m
\leq n. The measure R_m vanishes if the state is separable into m subsystems,
and for m = n it gives the Meyer-Wallach measure while for m = 2 it reduces, in
effect, to the one introduced recently by Love et al. The measures R_m are
evaluated explicitly for the GHZ state and the W state (and its modifications,
the W_k states) to show that these globally entangled states exhibit rather
distinct behaviors under the measures, indicating the utility of the measures
R_m for characterizing globally entangled states as well.Comment: 8 pages, 8 figure
Special Massive Spin-2 on de Sitter Space
The theory of a massive spin-2 state on the de Sitter space -- with the mass
squared equal to one sixth of the curvature -- is special for two reasons: (i)
it exhibits an enhanced local symmetry; (ii) it emerges as a part of the model
that gives rise to the self-accelerated Universe. The known problems of this
theory are: either it cannot be coupled to a non-conformal conserved
stress-tensor because of the enhanced symmetry, or it propagates a ghost-like
state when the symmetry is constrained by the Lagrange multiplier method. Here
we propose a solution to these problems in the linearized approximation.Comment: 9 pages, reference added, JCAP versio
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