Connectivity of random k-nearest-neighbour graphs

Let 𝓅 be a Poisson process of intensity one in a square S n of area n. We construct a random geometric graph G n,k by joining each point of 𝓅 to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that G n, k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that G n, k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With G n, k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover S n tends to 0 as n → ∞ while, if k ≥ 0.9967 log n, then the probability that the discs cover S n tends to 1 as n → ∞.

Connectivity of random k-nearest-neighbour graphs

Let 𝓅 be a Poisson process of intensity one in a square Sn of area n. We construct a random geometric graph Gn,k by joining each point of 𝓅 to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that Gn, k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that Gn, k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With Gn, k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover Sn tends to 0 as n → ∞ while, if k ≥ 0.9967 log n, then the probability that the discs cover Sn tends to 1 as n → ∞.

Assessment of Laminar-Turbulent Transition in Closed Disc Geometries

Finite-difference solutions are presented for rotationally-induced flows in the closed space between two coaxial discs and an outer cylindrical shroud, in which there is no superimposed flow. The solutions are obtained with an elliptic-flow calculation procedure and an anisotropic low turbulence Reynolds number k-ε model for the estimation of turbulent fluxes. The transition from laminar to turbulent flow is effected by including in the energy production term a small fraction (0.002) of the ‘turbulent viscosity’ as calculated from a simple mixing length model. This level for the artificial energy input was chosen as that appropriate for transition at a local rotational Reynolds number of 3 × 105 for the flow over a free, rotating disc. The main focus of the paper is the rotor-stator system, for which the influence of rotational Reynolds number (over the range 105–107) is investigated. Predicted velocity profiles and disc moment coefficients show reasonably good agreement with available experimental data. The computational procedure is then extended to cover the cases of corotating and counter-rotating systems, with variable relative disc speed.

A Lemma in complex function theory I

International audience Continuing our earlier work on the same topic published in the same journal last year we prove the following result in this paper: If $f(z)$ is analytic in the closed disc $\vert z\vert\leq r$ where $\vert f(z)\vert\leq M$ holds, and $A\geq1$, then $\vert f(0)\vert\leq(24A\log M) (\frac{1}{2r}\int_{-r}^r \vert f(iy)\vert\,dy)+M^{-A}.$ Proof uses an averaging technique involving the use of the exponential function and has many applications to Dirichlet series and the Riemann zeta function.

On zero-trace commutators

We present some results concerning the trace of certain trace class commutators of operators acting on a separable, complex Hilbert space. It is shown, among other things, that if X is a Hilbert-Schmidt operator and A is an operator such that AX − XA is a trace class operator, then tr (AX − XA) = 0 provided one of the following conditions holds: (a) A is subnormal and A*A − AA* is a trace class operator, (b) A is a hyponormal contraction and 1 − AA* is a trace class operator, (c) A2 is normal and A*A − AA* is a trace class operator, (d) A2 and A3 are normal. It is also shown that if A is a self - adjoint operator, if f is a function that is analytic on some neighbourhood of the closed disc{z: |z| ≥ ||A||}, and if X is a compact operator such that f (A) X − Xf (A) is a trace class operator, then tr (f (A) X − Xf (A))=0.

On the number of clumps resulting from the overlap of randomly placed figures in a plane

When two-dimensional figures, called laminae, are randomly placed on a plane domains result that can either be aggregates or individual laminae. The intersection of the union, U, of these domains with a specified field of view, F, in the plane is considered. The separate elements of the intersection are called clumps; they may be laminae, aggregates or partial laminae and aggregates. A formula is derived for the expected number of clumps minus enclosed voids. For bounded laminae homeomorphic to a closed disc with isotropic random direction the formula contains only their mean area and mean perimeter, the area and perimeter of F, and the intensity of the Poisson process.

On the number of clumps resulting from the overlap of randomly placed figures in a plane

When two-dimensional figures, called laminae, are randomly placed on a plane domains result that can either be aggregates or individual laminae. The intersection of the union, U, of these domains with a specified field of view, F, in the plane is considered. The separate elements of the intersection are called clumps; they may be laminae, aggregates or partial laminae and aggregates. A formula is derived for the expected number of clumps minus enclosed voids. For bounded laminae homeomorphic to a closed disc with isotropic random direction the formula contains only their mean area and mean perimeter, the area and perimeter of F, and the intensity of the Poisson process.

A conjecture of Fejes Tóth on saturated systems of circles

A family ℳ of closed circular discs in the plane is called a saturated family or a saturated system of circles if (i) the infimum r of the radii of the discs in ℳ is positive and (ii) every closed disc of radius r in the plane intersects at least one disc in ℳ. For a saturated family ℳ, we denote by S the point-set union of the interiors of the members of ℳ and by S(l) the part of S inside the circular disc of radius l centred at the origin. We define the lower density ρℳ of the saturated family ℳ aswhere V(S(l)) denotes the Lebesgue measure of the set S(l).