Let 𝒪_K be the ring of integers of an imaginary quadratic field K. Recently, Ji and Xie proved that every rational map f :ℂ→ℂ of degree d ≥ 2 whose multipliers all lie in 𝒪_K is a power map, a Chebyshev map or a Lattès map. Their proof relies on a result from non-Archimedean dynamics obtained by Rivera-Letelier. In the present note, we show that one can avoid using this result by considering a differential equation instead. Our proof of Ji and Xie’s result also applies to the case of entire maps. Thus, we also show that every nonaffine entire map f :ℂ→ℂ whose multipliers all lie in 𝒪_K is a power map or a Chebyshev map.
In this paper we investigate higher moments attached to the Chebotarev Density Theorem. Our focus is on the impact that peculiar Galois group structures have on the limiting distribution. Precisely we consider in this paper the case of groups having a character of large degree. Under the Generalized Riemann Hypothesis, we prove in particular that there exists families of Galois extensions of number fields having doubly transitive Frobenius group for which no Gaussian limiting distribution occurs.
This paper presents a Lattice Boltzmann Method (LBM) tailored for solving drift-diffusion equations in cold plasma applications. The proposed scheme is aimed to be a first step to address the challenges of simulating cold plasmas, characterized by non-equilibrium conditions and complex interactions among electrons, ions, and electric fields. By employing a parabolic scaling and simplifying assumptions, the method ensures computational efficiency while maintaining accuracy. Validation is performed through numerical test cases. While promising, future work will focus on incorporating energy dynamics and handling variable diffusion coefficients to enhance the method’s applicability in diverse plasma scenarios.
A k-wise B-divisible set family is a collection F of subsets of {1,. .., n} such that any intersection of k sets in F has cardinality divisible by B. If k = B = 2, it is well-known that F <= 2tn/21. We generalise this by proving that F <= 2tn/p1 if k = B = p, for any prime number p. For arbitrary values of B, we prove that 4B2-wise B-divisible set families F satisfy F <= 2tn/& ell;1 and that the only families achieving the upper bound are atomic, meaning that they consist of all the unions of disjoint subsets of size B. This improves upon a recent result by Gishboliner, Sudakov and Timon, that arrived at the same conclusion for k-wise Bdivisible families, with values of k that behave exponentially in B. Our techniques rely heavily upon a coding-theory analogue of Kneser's Theorem from additive combinatorics. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We consider the problem of Bayesian estimation of static parameters associated to a partially and discretely observed diffusion process. We assume that the exact transition dynamics of the diffusion process are unavailable, even up-to an unbiased estimator and that one must time-discretize the diffusion process. In such scenarios it has been shown how one can introduce the multilevel Monte Carlo method to reduce the cost to compute posterior expected values of the parameters for a pre-specified mean square error (MSE). These afore-mentioned methods rely on upon the Euler-Maruyama discretization scheme which is well-known in numerical analysis to have slow convergence properties. We adapt stochastic Runge-Kutta (SRK) methods for Bayesian parameter estimation of static parameters for diffusions. This can be implemented in high-dimensions of the diffusion and seemingly under-appreciated in the uncertainty quantification and statistics fields. For a class of diffusions and SRK methods, we consider the estimation of the posterior expectation of the parameters. We prove that to achieve a MSE of $\mathcal{O}(\epsilon^2)$, for $\epsilon>0$ given, the associated work is $\mathcal{O}(\epsilon^{-2})$. Whilst the latter is achievable for the Milstein scheme, this method is often not applicable for diffusions in dimension larger than two. We also illustrate our methodology in several numerical examples.
