572 research outputs found
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
Ionisation by quantised electromagnetic fields: The photoelectric effect
In this paper we explain the photoelectric effect in a variant of the
standard model of non relativistic quantum electrodynamics, which is in some
aspects more closely related to the physical picture, than the one studied in
[BKZ]: Now we can apply our results to an electron with more than one bound
state and to a larger class of electron-photon interactions. We will specify a
situation, where ionisation probability in second order is a weighted sum of
single photon terms. Furthermore we will see, that Einstein's equality
for the maximal kinetic energy of
the electron, energy of the photon and ionisation gap
is the crucial condition for these single photon terms to be nonzero.Comment: 59 pages, LATEX2
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]
Many parameter Hoelder perturbation of unbounded operators
If is a -mapping, for , having
as values unbounded self-adjoint operators with compact resolvents and common
domain of definition, parametrized by in an (even infinite dimensional)
space, then any continuous (in ) arrangement of the eigenvalues of is
indeed in .Comment: LaTeX, 4 pages; The result is generalized from Lipschitz to Hoelder.
Title change
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
Robots with Lights: Overcoming Obstructed Visibility Without Colliding
Robots with lights is a model of autonomous mobile computational entities
operating in the plane in Look-Compute-Move cycles: each agent has an
externally visible light which can assume colors from a fixed set; the lights
are persistent (i.e., the color is not erased at the end of a cycle), but
otherwise the agents are oblivious. The investigation of computability in this
model, initially suggested by Peleg, is under way, and several results have
been recently established. In these investigations, however, an agent is
assumed to be capable to see through another agent. In this paper we start the
study of computing when visibility is obstructable, and investigate the most
basic problem for this setting, Complete Visibility: The agents must reach
within finite time a configuration where they can all see each other and
terminate. We do not make any assumption on a-priori knowledge of the number of
agents, on rigidity of movements nor on chirality. The local coordinate system
of an agent may change at each activation. Also, by definition of lights, an
agent can communicate and remember only a constant number of bits in each
cycle. In spite of these weak conditions, we prove that Complete Visibility is
always solvable, even in the asynchronous setting, without collisions and using
a small constant number of colors. The proof is constructive. We also show how
to extend our protocol for Complete Visibility so that, with the same number of
colors, the agents solve the (non-uniform) Circle Formation problem with
obstructed visibility
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
Invariants of Artinian Gorenstein Algebras and Isolated Hypersurface Singularities
We survey our recently proposed method for constructing biholomorphic
invariants of quasihomogeneous isolated hypersurface singularities and, more
generally, invariants of graded Artinian Gorenstein algebras. The method
utilizes certain polynomials associated to such algebras, called
nil-polynomials, and we compare them with two other classes of polynomials that
have also been used to produce invariants.Comment: 13 page
Potential Conservation Laws
We prove that potential conservation laws have characteristics depending only
on local variables if and only if they are induced by local conservation laws.
Therefore, characteristics of pure potential conservation laws have to
essentially depend on potential variables. This statement provides a
significant generalization of results of the recent paper by Bluman, Cheviakov
and Ivanova [J. Math. Phys., 2006, V.47, 113505]. Moreover, we present
extensions to gauged potential systems, Abelian and general coverings and
general foliated systems of differential equations. An example illustrating
possible applications of proved statements is considered. A special version of
the Hadamard lemma for fiber bundles and the notions of weighted jet spaces are
proposed as new tools for the investigation of potential conservation laws.Comment: 36 pages, extended versio
On Z-gradations of twisted loop Lie algebras of complex simple Lie algebras
We define the twisted loop Lie algebra of a finite dimensional Lie algebra
as the Fr\'echet space of all twisted periodic smooth mappings
from to . Here the Lie algebra operation is
continuous. We call such Lie algebras Fr\'echet Lie algebras. We introduce the
notion of an integrable -gradation of a Fr\'echet Lie algebra, and
find all inequivalent integrable -gradations with finite dimensional
grading subspaces of twisted loop Lie algebras of complex simple Lie algebras.Comment: 26 page
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