We study the Boltzmann equation with external forces, not necessarily deriving from a potential, in the incompressible Navier–Stokes perturbative regime. On the torus, we establish Cauchy theories that are independent of the Knudsen number in Sobolev spaces. The existence is proved around a time-dependent Maxwellian that behaves like the global equilibrium both as time grows and as the Knudsen number decreases. We combine hypocoercive properties of linearized Boltzmann operators with linearization around a time-dependent Maxwellian that catches the fluctuations of the characteristics trajectories due to the presence of the force. This uniform theory is sufficiently robust to derive the incompressible Navier–Stokes–Fourier system with an external force from the Boltzmann equation. Neither smallness nor a time-decaying assumption is required for the external force, nor a gradient form, and we deal with general hard potential and cut-off Boltzmann kernels. As a by-product, the latest general theories for unit Knudsen number when the force is sufficiently small and decays in time are recovered.

To handle time series with complicated oscillatory structure, we propose a novel time-frequency (TF) analysis tool that fuses the short-time Fourier transform (STFT) and periodic transform (PT). As many time series oscillate with time-varying frequency, amplitude and nonsinusoidal oscillatory pattern, a direct application of PT or STFT might not be suitable. However, we show that by combining them in a proper way, we obtain a powerful TF analysis tool. We first combine the Ramanujan sums and $l_1$ penalization to implement the PT. We call the algorithm Ramanujan PT (RPT). The RPT is of its own interest for other applications, like analyzing short signals composed of components with integer periods, but that is not the focus of this paper. Second, the RPT is applied to modify the STFT and generate a novel TF representation of the complicated time series that faithfully reflects the instantaneous frequency information of each oscillatory component. We coin the proposed TF analysis the Ramanujan de-shape (RDS) and vectorized RDS (vRDS). In addition to showing some preliminary analysis results on complicated biomedical signals, we provide theoretical analysis about the RPT. Specifically, we show that the RPT is robust to three commonly encountered noises, including envelop fluctuation, jitter and additive noise.

This article concerns joint asymptotics of Fourier coefficients of restrictions of Laplace eigenfunctions ${\phi }_j$ of a compact Riemannian manifold to a submanifold $H\subset M$. We fix a number $c\in (0,1)$ and study the asymptotics of the thin sums,

where $\{{\lambda }_j\}$ are the eigenvalues of ${\sqrt{-\mathrm{\Delta }}}_M$, and $\{({\mu }_k,{\psi }_k)\}$ are the eigenvalues and the corresponding eigenfunctions of ${\sqrt{-\mathrm{\Delta }}}_H$. The inner sums represent the “jumps” of $N_{𝜖,H}^c(\lambda )$ and reflect the geometry of geodesic $c$-biangles with one leg on $H$ and a second leg on $M$ with the same endpoints and compatible initial tangent vectors $\xi \in S_H^{c}M$, ${\pi }_H\xi \in B^{\ast }H$, where ${\pi }_H\xi$ is the orthogonal projection of $\xi$ to $H$. A $c$-biangle occurs when $\left|{\pi }_H\xi \right|∕\left|\xi \right|=c$. Smoothed sums in ${\mu }_k$ are also studied and give sharp estimates on the jumps. The jumps themselves may jump as $𝜖$ varies, at certain values of $𝜖$ related to periodicities in the $c$-biangle geometry. Subspheres of spheres and certain subtori of tori illustrate these jumps. The results refine those of our previous article, where the inner sums run over $k$ such that $|{\mu }_k∕{\lambda }_j-c|\le 𝜖$ and where geodesic biangles do not play a role.

We consider the Dirac operator on asymptotically static Lorentzian manifolds with an odd-dimensional compact Cauchy surface. We prove that if Atiyah–Patodi–Singer boundary conditions are imposed at infinite times then the Dirac operator is Fredholm. This generalizes a theorem due to Bär and Strohmaier (Amer. J. Math. 141:5 (2019), 1421–1455) in the case of finite times, and we also show that the corresponding index formula extends to the infinite setting. Furthermore, we demonstrate the existence of a Fredholm inverse which is at the same time a Feynman parametrix in the sense of Duistermaat and Hörmander. The proof combines methods from time-dependent scattering theory with a variant of Egorov’s theorem for pseudodifferential hyperbolic systems.

We prove standard and reversed Strichartz estimates for the Klein–Gordon equation in ${ℝ}^{3+1}$. Instead of the Fourier theory, our analysis is based on fundamental solutions of the free equations and fractional integrations. We apply Strichartz estimates to study semilinear Klein–Gordon equations.