Stable divisorial gonality is in NP

Divisorial gonality and stable divisorial gonality are graph parameters, which have an origin in algebraic geometry. Divisorial gonality of a connected graph $G$ can be defined with help of a chip firing game on $G$. The stable divisorial gonality of $G$ is the minimum divisorial gonality over all subdivisions of edges of $G$. In this paper we prove that deciding whether a given connected graph has stable divisorial gonality at most a given integer $k$ belongs to the class NP. Combined with the result that (stable) divisorial gonality is NP-hard by Gijswijt, we obtain that stable divisorial gonality is NP-complete. The proof consist of a partial certificate that can be verified by solving an Integer Linear Programming instance. As a corollary, we have that the number of subdivisions needed for minimum stable divisorial gonality of a graph with $n$ vertices is bounded by $2^{p(n)}$ for a polynomial $p$

Correcting mean-field approximations for spatially-dependent advection-diffusion-reaction processes

On the microscale, migration, proliferation and death are crucial in the development, homeostasis and repair of an organism; on the macroscale, such effects are important in the sustainability of a population in its environment. Dependent on the relative rates of migration, proliferation and death, spatial heterogeneity may arise within an initially uniform field; this leads to the formation of spatial correlations and can have a negative impact upon population growth. Usually, such effects are neglected in modeling studies and simple phenomenological descriptions, such as the logistic model, are used to model population growth. In this work we outline some methods for analyzing exclusion processes which include agent proliferation, death and motility in two and three spatial dimensions with spatially homogeneous initial conditions. The mean-field description for these types of processes is of logistic form; we show that, under certain parameter conditions, such systems may display large deviations from the mean field, and suggest computationally tractable methods to correct the logistic-type description

Air speed and attitude probe

An air speed and attitude probe characterized by a pivot shaft normally projected from a data boom and supported thereby for rotation about an axis of rotation coincident with the longitudinal axis of the shaft is described. The probe is a tubular body supported for angular displacement about the axis of rotation and has a fin mounted on the body for maintaining one end of the body in facing relation with relative wind and has a pair of transducers mounted in the body for providing intelligence indicative of total pressure and static pressure for use in determining air speed. A stack of potentiometers coupled with the shaft to provide intelligence indicative of aircraft attitude, and circuitry connecting the transducers and potentiometers to suitable telemetry circuits are described

Models of collective cell motion for cell populations with different aspect ratio: diffusion, proliferation & travelling waves

Continuum, partial differential equation models are often used to describe the collective motion of cell populations, with various types of motility represented by the choice of diffusion coefficient, and cell proliferation captured by the source terms. Previously, the choice of diffusion coefficient has been largely arbitrary, with the decision to choose a particular linear or nonlinear form generally based on calibration arguments rather than making any physical connection with the underlying individual-level properties of the cell motility mechanism. In this work we provide a new link between individual-level models, which account for important cell properties such as varying cell shape and volume exclusion, and population-level partial differential equation models. We work in an exclusion process framework, considering aligned, elongated cells that may occupy more than one lattice site, in order to represent populations of agents with different sizes. Three different idealisations of the individual-level mechanism are proposed, and these are connected to three different partial differential equations, each with a different diffusion coefficient; one linear, one nonlinear and degenerate and one nonlinear and nondegenerate. We test the ability of these three models to predict the population-level response of a cell spreading problem for both proliferative and nonproliferative cases. We also explore the potential of our models to predict long time travelling wave invasion rates and extend our results to two-dimensional spreading and invasion. Our results show that each model can accurately predict density data for nonproliferative systems, but that only one does so for proliferative systems. Hence great care must be taken to predict density data with varying cell shape

Gravitational waves from black hole collisions via an eclectic approach

We present the first results in a new program intended to make the best use of all available technologies to provide an effective understanding of waves from inspiralling black hole binaries in time for imminent observations. In particular, we address the problem of combining the close-limit approximation describing ringing black holes and full numerical relativity, required for essentially nonlinear interactions. We demonstrate the effectiveness of our approach using general methods for a model problem, the head-on collision of black holes. Our method allows a more direct physical understanding of these collisions indicating clearly when non-linear methods are important. The success of this method supports our expectation that this unified approach will be able to provide astrophysically relevant results for black hole binaries in time to assist gravitational wave observations.Comment: 4 pages, 3 eps figures, Revte

Chiral-symmetry breaking in dual QCD

In the context of the formulation of QCD with dual potentials, we show that chiral-symmetry breaking occurs only in the confined state. Therefore, the transition temperature, beyond which chiral symmetry is restored, is the same as the deconfinement temperature. To carry out the calculation, it is necessary to couple quarks to dual gluons. We indicate how this is done (to lowest order in the magnetic coupling constant) and give the Feynman rules for quark–dual-gluon vertices

Static quark potential according to the dual-superconductor picture of QCD

We use the effective action describing long-range QCD, which predicts that QCD behaves as a dual superconductor, to derive the interaction energy between two heavy quarks as a function of separation. The dual-superconductor field equations are solved in an approximation in which the boundary between the superconducting vacuum and the region of normal vacuum surrounding the quarks is sharp. Further, non-Abelian effects are neglected. The resulting heavy-quark potential is linear in separation at large separation, and Coulomb-like at small separation. Overall it agrees very well with phenomenologically determined potentials

Quantized electric-flux-tube solutions to Yang-Mills theory

We suggest that long-distance Yang-Mills theory is more conveniently described in terms of electric rather than the customary magnetic vector potentials. On this basis we propose as an effective Lagrangian for this regime the most simple gauge-invariant (under the magnetic rather than electric gauge group) and Lorentz-invariant Lagrangian which yields a 1/q^4 gluon propagator in the Abelian limit. The resulting classical equations of motion have solutions corresponding to tubes of color electric flux quantized in units of e/2 (e is the Yang-Mills coupling constant). To exponential accuracy the electric color energy is contained in a cylinder of finite radius, showing that continuum Yang-Mills theory has excitations which are confined tubes of color electric flux. This is the criterion for electric confinement of color

A Constituent Quark Anti-Quark Effective Lagrangian Based on the Dual Superconducting Model of Long Distance QCD

We review the assumptions leading to the description of long distance QCD by a Lagrangian density expressed in terms of dual potentials. We find the color field distribution surrounding a quark anti-quark pair to first order in their velocities. Using these distributions we eliminate the dual potentials from the Lagrangian density and obtain an effective interaction Lagrangian $L_I ( \vec x_1 \, , \vec x_2 \, ; \vec v_1 \, , \vec v_2 )$ depending only upon the quark and anti-quark coordinates and velocities, valid to second order in their velocities. We propose $L_I$ as the Lagrangian describing the long distance interaction between constituent quarks. Elsewhere we have determined the two free parameters in $L_I$, $\alpha_s$ and the string tension $\sigma$, by fitting the 17 known levels of $b \bar b$ and $c \bar c$ systems. Here we use $L_I$ at the classical level to calculate the leading Regge trajectory. We obtain a trajectory which becomes linear at large $M^2$ with a slope $\alpha' \simeq .74 \, \hbox{GeV}^{-1}$, and for small $M^2$ the trajectory bends so that there are no tachyons. For a constituent quark mass between 100 and 150 MeV this trajectory passes through the two known Regge recurrences of the $\pi$ meson. In this paper, for simplicity of presentation, we have treated the quarks as spin-zero particles.Comment: {\bf 32,UW/PT94-0