In this work, we introduce multi-Novikov algebras, a generalisation of Novikov algebras with several binary operations indexed by a given set, and show that the multi-indices recently introduced in the context of singular stochastic partial differential equations can be interpreted as free multi-Novikov algebras. This is parallel to the fact that decorated rooted trees arising in the context of regularity structures are related to free multi-pre-Lie algebras.
We give a proof of the convergence of the BHZ renormalized model associated with the generalized (KPZ) equation that does not require the full strength of the BPHZ renormalisation. Our approach is based on a convenient form of chaos decomposition. The other key ingredient is a generalisation of the Hairer-Quastel convergence theorem for Feynman diagrams with certain decorations encoding Taylor remainders. With these ideas we are able to construct the model for the generalised KPZ equation.
In this paper we slightly improve the regularity theory for the so called optimal design problem. We first establish the uniform rectifiability of the boundary of the optimal set, for a larger class of minimizers, in any dimension. As an application, we improve the bound obtained by Larsen in dimension 2 about the mutual distance between two connected components. Finally we also prove that the full regularity in dimension 2 holds true provided that the ratio between the two constants in front of the Dirichlet energy is not larger than 4, which partially answers to a question raised by Larsen.
We study the problem of determining which integer polynomials divide Newman polynomials. In this vein, we first give results concerning the $8438$ known polynomials with Mahler measure less than $1.3$. We then exhibit a list of polynomials that divide no Newman polynomial. In particular, we show that a degree-10 polynomial of Mahler measure \text{approximately} 1.419404632 divides no Newman polynomial, thereby improving the best known upper bound for any universal constant $σ$, if it exists, such that every integer polynomial of Mahler measure less than $σ$ divides a Newman polynomial. Finally, letting $l(x)$ denote Lehmer's polynomial, we explicitly construct Newman polynomials divisible by $l(x)^2$ with degrees up to $150$, and show that no Newman polynomial is divisible by $l(x)^3$ up to degree $160$.
We study the minimization, under a volume constraint, of the first eigenvalue of the Lamé system in R N .We first prove the existence of an optimal domain in the class of quasi-open sets and show that, in the physically relevant dimensions N = 2 and N = 3, every optimal domain is open.We then derive first and second-order optimality conditions.In dimension two, these conditions imply that the disk is not optimal when the Poisson ratio is below a certain threshold, whereas it is a local minimizer above this threshold.We further prove that the disk is not optimal for all Poisson ratios ν satisfying ν ≤ 0.4.
