Singular SPDEs in domains with boundaries

We study spaces of modelled distributions with singular behaviour near the boundary of a domain that, in the context of the theory of regularity structures, allow one to give robust solution theories for singular stochastic PDEs with boundary conditions. The calculus of modelled distributions established in Hairer (Invent. Math. 198, 2014) is extended to this setting. We formulate and solve fixed point problems in these spaces with a class of kernels that is sufficiently large to cover in particular the Dirichlet and Neumann heat kernels. These results are then used to provide solution theories for the KPZ equation with Dirichlet and Neumann boundary conditions and for the 2D generalised parabolic Anderson model with Dirichlet boundary conditions. In the case of the KPZ equation with Neumann boundary conditions, we show that, depending on the class of mollifiers one considers, a "boundary renormalisation" takes place. In other words, there are situations in which a certain boundary condition is applied to an approximation to the KPZ equation, but the limiting process is the Hopf-Cole solution to the KPZ equation with a different boundary condition

Push sum with transmission failures

The push-sum algorithm allows distributed computing of the average on a directed graph, and is particularly relevant when one is restricted to one-way and/or asynchronous communications. We investigate its behavior in the presence of unreliable communication channels where messages can be lost. We show that exponential convergence still holds and deduce fundamental properties that implicitly describe the distribution of the final value obtained. We analyze the error of the final common value we get for the essential case of two nodes, both theoretically and numerically. We provide performance comparison with a standard consensus algorithm

On the interplay between Babai and Cerny's conjectures

Motivated by the Babai conjecture and the Cerny conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with $n$ states in this class, we prove that the reset thresholds are upper-bounded by $2n^2-6n+5$ and can attain the value $\tfrac{n(n-1)}{2}$. In addition, we study diameters of the pair digraphs of permutation automata and construct $n$-state permutation automata with diameter $\tfrac{n^2}{4} + o(n^2)$.Comment: 21 pages version with full proof

Trajectory convergence from coordinate-wise decrease of general convex energy functions

We consider arbitrary trajectories subject to a coordinate-wise energy decrease: the sign of the derivative of each entry is never the same as that of the corresponding entry of the gradient of some convex energy function. We show that this simple condition guarantees convergence to a point, to the minimum of the energy functions, or to a set where its Hessian has very specific properties. This extends and strengthens recent results that were restricted to quadratic energy functions

Effect of adult weight and CT-based selection on the performances of growing rabbits

The aim of the study was to compare the productive performance of different genotypes. Maternal (M; n=32, adult weight /AW/ 4.0-4.5kg, selected for number of kits born alive), Pannon White (P; n=32, AW: 4.3-4.8kg), and Large body line (L; n=32, AW: 4.8- 5.4kg) (P and L were selected for carcass traits based on CT /Computer tomography/data) rabbits were analysed. Average daily gain between 5-11wk of age, body weight at 11wk of age and feed intake were significantly (P<0.001) highest for L rabbits. For M, P and L rabbits, the following values were observed: average daily gain=38.6, 43.1 and 47.4g/d; body weight=2458, 2667 and 2949g; feed intake=115, 121 and 138g/d, respectively. Mortality of growing rabbits was unaffected by genotype. It can be concluded that production traits were mainly affected by the adult weight of the genotypes

Trajectory convergence from coordinate-wise decrease of quadratic energy functions, and applications to platoons

We consider trajectories where the sign of the derivative of each entry is opposite to that of the corresponding entry in the gradient of an energy function. We show that this condition guarantees convergence when the energy function is quadratic and positive definite and partly extend that result to some classes of positive semi-definite quadratic functions including those defined using a graph Laplacian. We show how this condition allows establishing the convergence of a platoon application in which it naturally appears, due to deadzones in the control laws designed to avoid instabilities caused by inconsistent measurements of the same distance by different agents