Tutte′s 3-Flow Conjecture and Short Cycle Covers

AbstractIn this paper we prove: (i) If a graph G has a nowhere-zero 6-flow φ such that | Eodd (φ)| ≥ 23 |E(G)|, then G has a cycle cover in which the sum of the lengths of the cycles in the cycle cover is at most 4427 |E(G)|, where Eodd (φ) = {e ∈ E(G) : is odd}; (ii) if Tutte′s 3-Flow Conjecture is true, then every bridgeless graph G has a nowhere-zero 6-flow φ such that | Eodd (φ)| ≥ 23 | E(G)|

Fulkerson′s Conjecture and Circuit Covers

AbstractIt was conjectured by Fulkerson that the edge-set of any bridgeless graph can be covered by six cycles (union of circuits) such that each edge is in exactly four cycles. We prove that if Fulkerson′s conjecture is true, then the edge-set of every bridgeless graph G can be covered by three cycles whose total length is at most 2215|E(G)|. We also prove that there are infinitely many bridgeless graphs G whose edge-set cannot be covered by three cycles of total length less than 2215|E(G)|

Diffusion as mixing mechanism in granular materials

We present several numerical results on granular mixtures. In particular, we examine the efficiency of diffusion as a mixing mechanism in these systems. The collisions are inelastic and to compensate the energy loss, we thermalize the grains by adding a random force. Starting with a segregated system, we show that uniform agitation (heating) leads to a uniform mixture of grains of different sizes. We define a characteristic mixing time, $\tau_{mix}$, and study theoretically and numerically its dependence on other parameters like the density. We examine a model for bidisperse systems for which we can calculate some physical quantities. We also examine the effect of a temperature gradient and demonstrate the appearance of an expected segregation.Comment: 15 eps figures, include

Notes on entropic characteristics of quantum channels

One of most important issues in quantum information theory concerns transmission of information through noisy quantum channels. We discuss few channel characteristics expressed by means of generalized entropies. Such characteristics can often be dealt in line with more usual treatment based on the von Neumann entropies. For any channel, we show that the $q$-average output entropy of degree $q\geq1$ is bounded from above by the $q$-entropy of the input density matrix. Concavity properties of the $(q,s)$-entropy exchange are considered. Fano type quantum bounds on the $(q,s)$-entropy exchange are derived. We also give upper bounds on the map $(q,s)$-entropies in terms of the output entropy, corresponding to the completely mixed input.Comment: 10 pages, no figures. The statement of Proposition 1 is explicitly illustrated with the depolarizing channel. The bibliography is extended and updated. More explanations. To be published in Cent. Eur. J. Phy

Local linear regression with adaptive orthogonal fitting for the wind power application

Short-term forecasting of wind generation requires a model of the function for the conversion of me-teorological variables (mainly wind speed) to power production. Such a power curve is nonlinear and bounded, in addition to being nonstationary. Local linear regression is an appealing nonparametric ap-proach for power curve estimation, for which the model coefficients can be tracked with recursive Least Squares (LS) methods. This may lead to an inaccurate estimate of the true power curve, owing to the assumption that a noise component is present on the response variable axis only. Therefore, this assump-tion is relaxed here, by describing a local linear regression with orthogonal fit. Local linear coefficients are defined as those which minimize a weighted Total Least Squares (TLS) criterion. An adaptive es-timation method is introduced in order to accommodate nonstationarity. This has the additional benefit of lowering the computational costs of updating local coefficients every time new observations become available. The estimation method is based on tracking the left-most eigenvector of the augmented covari-ance matrix. A robustification of the estimation method is also proposed. Simulations on semi-artificial datasets (for which the true power curve is available) underline the properties of the proposed regression and related estimation methods. An important result is the significantly higher ability of local polynomia

Narrow linewidth hybrid InP-TriPleX photonic integrated tunable laser based on silicon nitride micro-ring resonators

Detailed characterization of a hybrid integrated tunable laser based on micro-ring resonators shows a tuning range of 50 nm with ~40 kHz linewidth. The device demonstrates performance comparable with commercial external cavity lasers in 16QAM coherent system

Relations for certain symmetric norms and anti-norms before and after partial trace

Changes of some unitarily invariant norms and anti-norms under the operation of partial trace are examined. The norms considered form a two-parametric family, including both the Ky Fan and Schatten norms as particular cases. The obtained results concern operators acting on the tensor product of two finite-dimensional Hilbert spaces. For any such operator, we obtain upper bounds on norms of its partial trace in terms of the corresponding dimensionality and norms of this operator. Similar inequalities, but in the opposite direction, are obtained for certain anti-norms of positive matrices. Through the Stinespring representation, the results are put in the context of trace-preserving completely positive maps. We also derive inequalities between the unified entropies of a composite quantum system and one of its subsystems, where traced-out dimensionality is involved as well.Comment: 11 pages, no figures. A typo error in Eq. (5.15) is corrected. Minor improvements. J. Stat. Phys. (in press

Reflection and Ducting of Gravity Waves Inside the Sun

Internal gravity waves excited by overshoot at the bottom of the convection zone can be influenced by rotation and by the strong toroidal magnetic field that is likely to be present in the solar tachocline. Using a simple Cartesian model, we show how waves with a vertical component of propagation can be reflected when traveling through a layer containing a horizontal magnetic field with a strength that varies with depth. This interaction can prevent a portion of the downward-traveling wave energy flux from reaching the deep solar interior. If a highly reflecting magnetized layer is located some distance below the convection zone base, a duct or wave guide can be set up, wherein vertical propagation is restricted by successive reflections at the upper and lower boundaries. The presence of both upward- and downward-traveling disturbances inside the duct leads to the existence of a set of horizontally propagating modes that have significantly enhanced amplitudes. We point out that the helical structure of these waves makes them capable of generating an alpha-effect, and briefly consider the possibility that propagation in a shear of sufficient strength could lead to instability, the result of wave growth due to over-reflection.Comment: 23 pages, 5 figures. Accepted for publication in Solar Physic