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 $a/b > 1$ 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 $a/b$ and use representations w.r.t. this base to construct normal numbers in base $a$ 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 $\mathbb{A}_\mathbb{Q}$ and Fourier analysis in $\mathbb{A}_\mathbb{Q}$. 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 ${}^{88}$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 $F=0.940(5)$ at an average rate $182\,\mathrm{s}^{-1}$ (success probability $2.18\times10^{-4}$).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 $A$, $B$, and $C$. The system is described by a grand canonical ensemble with temperature $T$ and chemical potentials $T\lambda_A$, $T\lambda_B$, and $T\lambda_C$. We find that for $\lambda_A=\lambda_B=\lambda_C$ the system undergoes a phase transition from a uniform density to a continuum of phases at a critical temperature $\hat T_c=(2\pi/\sqrt3)^{-1}$. For other values of the chemical potentials the system has a unique equilibrium state. As is the case for the canonical ensemble for this $ABC$ model, the grand canonical ensemble is the stationary measure satisfying detailed balance for a natural dynamics. We note that $\hat T_c=3T_c$, where $T_c$ is the critical temperature for a similar transition in the canonical ensemble at fixed equal densities $r_A=r_B=r_C=1/3$.Comment: 24 pages, 3 figure

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

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 $^{88}$Sr$^+$ ions in a $\lambda=674$ nm standing wave, whose relative position is controlled to $\approx\lambda/100$, we suppress the carrier coupling by a factor of $18$, 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