192 research outputs found
Modelling silicosis : structure of equilibria
We analyse the structure of equilibria of a coagulation–fragmentation–death model ofsilicosis. We present exact multiplicity results in the particular case of piecewise-constantcoefficients, results on existence and non-existence of equilibria in the general case, as wellas precise asymptotics for the infinite series that arise in the case of power law coefficient
Characterizing Triviality of the Exponent Lattice of A Polynomial through Galois and Galois-Like Groups
The problem of computing \emph{the exponent lattice} which consists of all
the multiplicative relations between the roots of a univariate polynomial has
drawn much attention in the field of computer algebra. As is known, almost all
irreducible polynomials with integer coefficients have only trivial exponent
lattices. However, the algorithms in the literature have difficulty in proving
such triviality for a generic polynomial. In this paper, the relations between
the Galois group (respectively, \emph{the Galois-like groups}) and the
triviality of the exponent lattice of a polynomial are investigated. The
\bbbq\emph{-trivial} pairs, which are at the heart of the relations between
the Galois group and the triviality of the exponent lattice of a polynomial,
are characterized. An effective algorithm is developed to recognize these
pairs. Based on this, a new algorithm is designed to prove the triviality of
the exponent lattice of a generic irreducible polynomial, which considerably
improves a state-of-the-art algorithm of the same type when the polynomial
degree becomes larger. In addition, the concept of the Galois-like groups of a
polynomial is introduced. Some properties of the Galois-like groups are proved
and, more importantly, a sufficient and necessary condition is given for a
polynomial (which is not necessarily irreducible) to have trivial exponent
lattice.Comment: 19 pages,2 figure
Cutting edges at random in large recursive trees
We comment on old and new results related to the destruction of a random
recursive tree (RRT), in which its edges are cut one after the other in a
uniform random order. In particular, we study the number of steps needed to
isolate or disconnect certain distinguished vertices when the size of the tree
tends to infinity. New probabilistic explanations are given in terms of the
so-called cut-tree and the tree of component sizes, which both encode different
aspects of the destruction process. Finally, we establish the connection to
Bernoulli bond percolation on large RRT's and present recent results on the
cluster sizes in the supercritical regime.Comment: 29 pages, 3 figure
Patterns in rational base number systems
Number systems with a rational number as base have gained interest
in recent years. In particular, relations to Mahler's 3/2-problem as well as
the Josephus problem have been established. In the present paper we show that
the patterns of digits in the representations of positive integers in such a
number system are uniformly distributed. We study the sum-of-digits function of
number systems with rational base and use representations w.r.t. this
base to construct normal numbers in base in the spirit of Champernowne. The
main challenge in our proofs comes from the fact that the language of the
representations of integers in these number systems is not context-free. The
intricacy of this language makes it impossible to prove our results along
classical lines. In particular, we use self-affine tiles that are defined in
certain subrings of the ad\'ele ring and Fourier
analysis in . With help of these tools we are able to
reformulate our results as estimation problems for character sums
The power of choice in network growth
The "power of choice" has been shown to radically alter the behavior of a
number of randomized algorithms. Here we explore the effects of choice on
models of tree and network growth. In our models each new node has k randomly
chosen contacts, where k > 1 is a constant. It then attaches to whichever one
of these contacts is most desirable in some sense, such as its distance from
the root or its degree. Even when the new node has just two choices, i.e., when
k=2, the resulting network can be very different from a random graph or tree.
For instance, if the new node attaches to the contact which is closest to the
root of the tree, the distribution of depths changes from Poisson to a
traveling wave solution. If the new node attaches to the contact with the
smallest degree, the degree distribution is closer to uniform than in a random
graph, so that with high probability there are no nodes in the network with
degree greater than O(log log N). Finally, if the new node attaches to the
contact with the largest degree, we find that the degree distribution is a
power law with exponent -1 up to degrees roughly equal to k, with an
exponential cutoff beyond that; thus, in this case, we need k >> 1 to see a
power law over a wide range of degrees.Comment: 9 pages, 4 figure
High-rate, high-fidelity entanglement of qubits across an elementary quantum network
We demonstrate remote entanglement of trapped-ion qubits via a
quantum-optical fiber link with fidelity and rate approaching those of local
operations. Two Sr qubits are entangled via the polarization
degree of freedom of two photons which are coupled by high-numerical-aperture
lenses into single-mode optical fibers and interfere on a beamsplitter. A novel
geometry allows high-efficiency photon collection while maintaining unit
fidelity for ion-photon entanglement. We generate remote Bell pairs with
fidelity at an average rate (success
probability ).Comment: v2 updated to include responses to reviewers, as published in PR
The grand canonical ABC model: a reflection asymmetric mean field Potts model
We investigate the phase diagram of a three-component system of particles on
a one-dimensional filled lattice, or equivalently of a one-dimensional
three-state Potts model, with reflection asymmetric mean field interactions.
The three types of particles are designated as , , and . The system is
described by a grand canonical ensemble with temperature and chemical
potentials , , and . We find that for
the system undergoes a phase transition from a
uniform density to a continuum of phases at a critical temperature . For other values of the chemical potentials the system
has a unique equilibrium state. As is the case for the canonical ensemble for
this model, the grand canonical ensemble is the stationary measure
satisfying detailed balance for a natural dynamics. We note that , where is the critical temperature for a similar transition in
the canonical ensemble at fixed equal densities .Comment: 24 pages, 3 figure
Breaking the entangling gate speed limit for trapped-ion qubits using a phase-stable standing wave
All laser-driven entangling operations for trapped-ion qubits have hitherto
been performed without control of the optical phase of the light field, which
precludes independent tuning of the carrier and motional coupling. By placing
Sr ions in a nm standing wave, whose relative position
is controlled to , we suppress the carrier coupling by a
factor of , while coherently enhancing the spin-motion coupling. We
experimentally demonstrate that the off-resonant carrier coupling imposes a
speed limit for conventional traveling-wave M{\o}lmer-S{\o}rensen gates; we use
the standing wave to surpass this limit and achieve a gate duration of $15\
\mu$s, restricted by the available laser power.Comment: S. Saner and O. B\u{a}z\u{a}van contributed equally to this wor
The Supremum Norm of the Discrepancy Function: Recent Results and Connections
A great challenge in the analysis of the discrepancy function D_N is to
obtain universal lower bounds on the L-infty norm of D_N in dimensions d \geq
3. It follows from the average case bound of Klaus Roth that the L-infty norm
of D_N is at least (log N) ^{(d-1)/2}. It is conjectured that the L-infty bound
is significantly larger, but the only definitive result is that of Wolfgang
Schmidt in dimension d=2. Partial improvements of the Roth exponent (d-1)/2 in
higher dimensions have been established by the authors and Armen Vagharshakyan.
We survey these results, the underlying methods, and some of their connections
to other subjects in probability, approximation theory, and analysis.Comment: 15 pages, 3 Figures. Reports on talks presented by the authors at the
10th international conference on Monte Carlo and Quasi-Monte Carlo Methods in
Scientific Computing, Sydney Australia, February 2011. v2: Comments of the
referee are incorporate
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