Cumulus Media

We show that submultiplicative norms on section rings of polarised projective manifolds are asymptotically equivalent to sup-norms associated with metrics on the polarisation. We then discuss some applications to the spectral theory of submultiplicative filtrations, the asymptotic study of the Narasimhan-Simha pseudonorms, and holomorphic extension theorem. As an unexpected byproduct, we show that injective and projective tensor norms on symmetric algebras of finite dimensional complex normed vector spaces are asymptotically equivalent.

Recently, new models of the boundary condition for the Boltzmann equation were proposed on the basis of a kinetic model of gas-surface interactions [K. Aoki et al., Phys. Rev. E 106, 035306 (2022)]. In the present paper, the kernel representations of the models are given, and the models are applied to some basic problems of a rarefied gas between two parallel plates. To be more specific, the heat transfer between the plates with different temperatures, plane Couette flow, and plane Poiseuille flow driven by an external force are numerically investigated by using the Bhatnagar-Gross-Krook model of the Boltzmann equation and the new models of the boundary condition. The results are compared with those based on the conventional Maxwell-type boundary condition. It is shown that when the interaction of gas and solid molecules is strong, the results based on the new models tend to approach those based on the diffuse reflection. However, when the interaction is not strong, the former results deviate from those based on the Maxwell-type condition with a constant accommodation coefficient.

In 1986, Kato and Kuzumaki introduced some Diophantine properties of fields, called the C_i^q properties, and they hoped they would provide a good characterization of the cohomological dimension of fields. In this paper, we study the stability of some variants of the C_i^q properties under transcendental and algebraic extensions. As an application, we obtain the C_n^1 property for the field 𝐅_p(x_1,⋯,x_n).

Consider a sequence of $C^4$ Lorentzian metrics $\{h_n\}_{n=1}^{+\infty}$ on a manifold $\mathcal M$ satisfying the Einstein vacuum equation $\mathrm{Ric}(h_n)=0$. Suppose there exists a smooth Lorentzian metric $h_0$ on $\mathcal M$ such that $h_n\to h_0$ uniformly on compact sets. Assume also that on any compact set $K\subset \mathcal M$, there is a decreasing sequence of positive numbers $\lambda_n \to 0$ such that $$\|\partial^{\alpha} (h_n - h_0)\|_{L^{\infty}(K)} \lesssim \lambda_n^{1-|\alpha|},\quad |\alpha|\geq 4.$$ It is well-known that $h_0$, which represents a "high-frequency limit", is not necessarily a solution to the Einstein vacuum equation. Nevertheless, Burnett conjectured that $h_0$ must be isometric to a solution to the Einstein-massless Vlasov system. In this paper, we prove Burnett's conjecture assuming that $\{h_n\}_{n=1}^{+\infty}$ and $h_0$ in addition admit a $\mathbb U(1)$ symmetry and obey an elliptic gauge condition. The proof uses microlocal defect measures - we identify an appropriately defined microlocal defect measure to be the Vlasov measure of the limit spacetime. In order to show that this measure indeed obeys the Vlasov equation, we need some special cancellations which rely on the precise structure of the Einstein equations. These cancellations are related to a new "trilinear compensated compactness" phenomenon for solutions to (semilinear) elliptic and (quasilinear) hyperbolic equations.

We consider a model of the Riemann zeta function on the critical axis and study its maximum over intervals of length (log T )θ, where θ is either fixed or tends to zero at a suitable rate. It is shown that the deterministic level of the maximum interpolates smoothly between the ones of log-correlated variables and of i.i.d. random variables, exhibiting a smooth transition ‘from 3 4 to 1 4 ’ in the second order. This provides a natural context where extreme value statistics of log-correlated variables with time-dependent variance and rate occur. A key ingredient of the proof is a precise upper tail tightness estimate for the maximum of the model on intervals of size one, that includes a Gaussian correction. This correction is expected to be present for the Riemann zeta function and pertains to the question of the correct order of the maximum of the zeta function in large intervals.