528 research outputs found
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)
Deterministic meeting of sniffing agents in the plane
Two mobile agents, starting at arbitrary, possibly different times from
arbitrary locations in the plane, have to meet. Agents are modeled as discs of
diameter 1, and meeting occurs when these discs touch. Agents have different
labels which are integers from the set of 0 to L-1. Each agent knows L and
knows its own label, but not the label of the other agent. Agents are equipped
with compasses and have synchronized clocks. They make a series of moves. Each
move specifies the direction and the duration of moving. This includes a null
move which consists in staying inert for some time, or forever. In a non-null
move agents travel at the same constant speed, normalized to 1. We assume that
agents have sensors enabling them to estimate the distance from the other agent
(defined as the distance between centers of discs), but not the direction
towards it. We consider two models of estimation. In both models an agent reads
its sensor at the moment of its appearance in the plane and then at the end of
each move. This reading (together with the previous ones) determines the
decision concerning the next move. In both models the reading of the sensor
tells the agent if the other agent is already present. Moreover, in the
monotone model, each agent can find out, for any two readings in moments t1 and
t2, whether the distance from the other agent at time t1 was smaller, equal or
larger than at time t2. In the weaker binary model, each agent can find out, at
any reading, whether it is at distance less than \r{ho} or at distance at least
\r{ho} from the other agent, for some real \r{ho} > 1 unknown to them. Such
distance estimation mechanism can be implemented, e.g., using chemical sensors.
Each agent emits some chemical substance (scent), and the sensor of the other
agent detects it, i.e., sniffs. The intensity of the scent decreases with the
distance.Comment: A preliminary version of this paper appeared in the Proc. 23rd
International Colloquium on Structural Information and Communication
Complexity (SIROCCO 2016), LNCS 998
Some algebraic properties of differential operators
First, we study the subskewfield of rational pseudodifferential operators
over a differential field K generated in the skewfield of pseudodifferential
operators over K by the subalgebra of all differential operators.
Second, we show that the Dieudonne' determinant of a matrix
pseudodifferential operator with coefficients in a differential subring A of K
lies in the integral closure of A in K, and we give an example of a 2x2 matrix
differential operator with coefficients in A whose Dieudonne' determiant does
not lie in A.Comment: 15 page
Rational matrix pseudodifferential operators
The skewfield K(d) of rational pseudodifferential operators over a
differential field K is the skewfield of fractions of the algebra of
differential operators K[d]. In our previous paper we showed that any H from
K(d) has a minimal fractional decomposition H=AB^(-1), where A,B are elements
of K[d], B is non-zero, and any common right divisor of A and B is a non-zero
element of K. Moreover, any right fractional decomposition of H is obtained by
multiplying A and B on the right by the same non-zero element of K[d]. In the
present paper we study the ring M_n(K(d)) of nxn matrices over the skewfield
K(d). We show that similarly, any H from M_n(K(d)) has a minimal fractional
decomposition H=AB^(-1), where A,B are elements of M_n(K[d]), B is
non-degenerate, and any common right divisor of A and B is an invertible
element of the ring M_n(K[d]). Moreover, any right fractional decomposition of
H is obtained by multiplying A and B on the right by the same non-degenerate
element of M_n(K [d]). We give several equivalent definitions of the minimal
fractional decomposition. These results are applied to the study of maximal
isotropicity property, used in the theory of Dirac structures.Comment: 20 page
The locally covariant Dirac field
We describe the free Dirac field in a four dimensional spacetime as a locally
covariant quantum field theory in the sense of Brunetti, Fredenhagen and Verch,
using a representation independent construction. The freedom in the geometric
constructions involved can be encoded in terms of the cohomology of the
category of spin spacetimes. If we restrict ourselves to the observable algebra
the cohomological obstructions vanish and the theory is unique. We establish
some basic properties of the theory and discuss the class of Hadamard states,
filling some technical gaps in the literature. Finally we show that the
relative Cauchy evolution yields commutators with the stress-energy-momentum
tensor, as in the scalar field case.Comment: 36 pages; v2 minor changes, typos corrected, updated references and
acknowledgement
Measurement of time--varying Multiple--Input Multiple--Output Channels
We derive a criterion on the measurability / identifiability of
Multiple--Input Multiple--Output (MIMO) channels based on the size of the
so-called spreading support of its subchannels. Novel MIMO transmission
techniques provide high-capacity communication channels in time-varying
environments and exact knowledge of the transmission channel operator is of key
importance when trying to transmit information at a rate close to channel
capacity
Polarizations and differential calculus in affine spaces
Within the framework of mappings between affine spaces, the notion of -th
polarization of a function will lead to an intrinsic characterization of
polynomial functions. We prove that the characteristic features of derivations,
such as linearity, iterability, Leibniz and chain rules, are shared -- at the
finite level -- by the polarization operators. We give these results by means
of explicit general formulae, which are valid at any order , and are based
on combinatorial identities. The infinitesimal limits of the -th
polarizations of a function will yield its -th derivatives (without
resorting to the usual recursive definition), and the above mentioned
properties will be recovered directly in the limit. Polynomial functions will
allow us to produce a coordinate free version of Taylor's formula
Antilinear deformations of Coxeter groups, an application to Calogero models
We construct complex root spaces remaining invariant under antilinear
involutions related to all Coxeter groups. We provide two alternative
constructions: One is based on deformations of factors of the Coxeter element
and the other based on the deformation of the longest element of the Coxeter
group. Motivated by the fact that non-Hermitian Hamiltonians admitting an
antilinear symmetry may be used to define consistent quantum mechanical systems
with real discrete energy spectra, we subsequently employ our constructions to
formulate deformations of Coxeter models remaining invariant under these
extended Coxeter groups. We provide explicit and generic solutions for the
Schroedinger equation of these models for the eigenenergies and corresponding
wavefunctions. A new feature of these novel models is that when compared with
the undeformed case their solutions are usually no longer singular for an
exchange of an amount of particles less than the dimension of the
representation space of the roots. The simultaneous scattering of all particles
in the model leads to anyonic exchange factors for processes which have no
analogue in the undeformed case.Comment: 32 page
Perturbation Theory around Non-Nested Fermi Surfaces I. Keeping the Fermi Surface Fixed
The perturbation expansion for a general class of many-fermion systems with a
non-nested, non-spherical Fermi surface is renormalized to all orders. In the
limit as the infrared cutoff is removed, the counterterms converge to a finite
limit which is differentiable in the band structure. The map from the
renormalized to the bare band structure is shown to be locally injective. A new
classification of graphs as overlapping or non-overlapping is given, and
improved power counting bounds are derived from it. They imply that the only
subgraphs that can generate factorials in the order of the
renormalized perturbation series are indeed the ladder graphs and thus give a
precise sense to the statement that `ladders are the most divergent diagrams'.
Our results apply directly to the Hubbard model at any filling except for
half-filling. The half-filled Hubbard model is treated in another place.Comment: plain TeX with postscript figures in a uuencoded gz-compressed tar
file. Put it on a separate directory before unpacking, since it contains
about 40 files. If you have problems, requests or comments, send e-mail to
[email protected]
Want to Gather? No Need to Chatter!
A team of mobile agents, starting from different nodes of an unknown network,
possibly at different times, have to meet at the same node and declare that
they have all met. Agents have different labels and move in synchronous rounds
along links of the network. The above task is known as gathering and was
traditionally considered under the assumption that when some agents are at the
same node then they can talk. In this paper we ask the question of whether this
ability of talking is needed for gathering. The answer turns out to be no.
Our main contribution are two deterministic algorithms that always accomplish
gathering in a much weaker model. We only assume that at any time an agent
knows how many agents are at the node that it currently occupies but agents do
not see the labels of other co-located agents and cannot exchange any
information with them. They also do not see other nodes than the current one.
Our first algorithm works under the assumption that agents know a priori some
upper bound N on the network size, and it works in time polynomial in N and in
the length l of the smallest label. Our second algorithm does not assume any a
priori knowledge about the network but its complexity is exponential in the
network size and in the labels of agents. Its purpose is to show feasibility of
gathering under this harsher scenario.
As a by-product of our techniques we obtain, in the same weak model, the
solution of the fundamental problem of leader election among agents. As an
application of our result we also solve, in the same model, the well-known
gossiping problem: if each agent has a message at the beginning, we show how to
make all messages known to all agents, even without any a priori knowledge
about the network. If agents know an upper bound N on the network size then our
gossiping algorithm works in time polynomial in N, in l and in the length of
the largest message
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