In this paper, we show that “$L$-complete null hypersurfaces” (i.e. ruled null hypersurfaces foliated by entirety of light-like lines) as wave fronts in $(n+1)$-dimensional Lorentz–Minkowski space are canonically induced by hypersurfaces in $n$-dimensional Euclidean space. As an application, we show that most of null wave fronts can be realized as restrictions of certain $L$-complete null wave fronts. Moreover, we determine $L$-complete null wave fronts whose singular sets are compact.

Every $a$ in the torus $A = \mathbb{R}^{3}/2\mathbb{Z}^{3}$ determines a unique spherical, Euclidean or hyperbolic triangle $T(a)$ with angles $(\pi a_{i})$. In this paper we study the Galois orbits $\operatorname{Gal}(a)$ of torsion points $a \in A$, focusing on the ramification density $$ \rho(a) = \frac{|\{b \in \operatorname{Gal}(a) \: : \: T(b) \;\text{is spherical}\}|}{|\operatorname{Gal}(a)|} \cdot $$ We show that the closure $\overline{R}$ of the set of values of $\rho(a)$ is a countable subset of $[0, 1]$, with 0 and 1 as isolated points. The spectral gaps at 0 and 1 lead to general finiteness statements for the classical triangle groups $\Delta(p,q,r) \subset \operatorname{SL}_{2}(\mathbb{R})$. For example, we obtain a conceptual proof, based on equidistribution, that the set of arithmetic triangle groups is finite.

A minimal group action has essential holonomy if the set of points with non-trivial holonomy has positive measure. If such an action is topologically free, then having essential holonomy is equivalent to the action not being essentially free, which means that the set of points with non-trivial stabilizer has positive measure. In this paper, we investigate the relation between the property of having essential holonomy and structure of the acting group for minimal equicontinuous actions on Cantor sets. We show that if such a group action is locally quasi-analytic and has essential holonomy, then every commutator subgroup in the group lower central series has elements with positive measure set of points with non-trivial holonomy. In particular, we prove that a minimal equicontinuous Cantor action by a nilpotent group has no essential holonomy. We also show that the property of having essential holonomy is preserved under return equivalence and continuous orbit equivalence of minimal equicontinuous Cantor actions. Finally, we give examples to show that the assumption on the action that it is locally quasi-analytic is necessary.

Let $S$ be a non-uniruled (i.e., non-birationally ruled) smooth projective surface. We show that the tangent bundle $T_{S}$ is pseudo-effective if and only if the canonical divisor $K_{S}$ is nef and the second Chern class vanishes, i.e., $c_{2}(S) = 0$. Moreover, we study the blow-up of a non-rational ruled surface with pseudo-effective tangent bundle.

We study an incompressible viscous flow around an obstacle with an oscillating boundary that moves by a translational periodic motion, and we show existence of strong time-periodic solutions for small data in different configurations: If the mean velocity of the body is zero, existence of time-periodic solutions is provided within a framework of Sobolev functions with isotropic pointwise decay. If the mean velocity is non-zero, this framework can be adapted, but the spatial behavior of the flow requires a setting of anisotropically weighted spaces. In the latter case, we also establish existence of solutions within an alternative framework of homogeneous Sobolev spaces. These results are based on the time-periodic maximal regularity of the associated linearizations, which is derived from suitable $\mathscr{R}$-bounds for the Stokes and Oseen resolvent problems. The pointwise estimates are deduced from the associated time-periodic fundamental solutions.

We consider a generalized Gambaudo–Ghys construction on bounded cohomology and prove its injectivity. As a corollary, we prove that the third bounded cohomology of the group of area-preserving diffeomorphisms on the 2-disk is infinite-dimensional. Additionally, we establish similar results for the 2-sphere, the 2-torus, and the annulus.

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$, and $M$ a finitely generated $R$-module. The asymptotic behavior of the quotient modules $M/I^{n} M$ of $M$ is an actively studied subject in commutative algebra. The main result of this paper shows that for large integers $n > 0$, the depth of the localizations of $(M/I^{n} M)_{\mathfrak{p}}$ are stable uniformly for all prime ideals $\mathfrak{p}$ of $R$ in each of the following cases: (1) $R$ is CM-excellent, (2) $R$ is semi-local, (3) $M$ or $M/I^{n} M$ for some $n > 0$ is Cohen–Macaulay.

In this note, we establish Guan–Zhou's unified version of optimal $L^{2}$ extension theorem for holomorphic vector bundles with smooth Hermitian metrics on weakly pseudoconvex Kähler manifolds. Combining with previous work of Guan–Mi–Yuan, we generalize Guan–Zhou's unified version of optimal $L^{2}$ extension theorem to weakly pseudoconvex Kähler manifolds.

It was conjectured that there are no hyperharmonic integers $h_{n}^{(r)}$ except 1. In 2020, a disproof of this conjecture was given by showing the existence of infinitely many hyperharmonic integers. However, the corresponding proof does not give any general density results related to hyperharmonic integers. In this paper, we first get better error estimates for the counting function of the pairs $(n, r)$ that correspond to non-integer hyperharmonic numbers using sums on gaps between consecutive prime numbers. Then, based on a plausible assumption on prime powers with restricted digits, we show that there exist positive integers $n$ such that the set of positive integers $r$ where $h_{n}^{(r)} \in \mathbb{Z}$ has positive density. Apart from that, we also obtain exact densities of sets $\{ r \in \mathbb Z_{>0} : h_{33}^{(r)} \in \mathbb{Z} \}$ and $\{ r \in \mathbb Z_{>0} : h_{39}^{(r)} \in \mathbb{Z} \}$. Finally, we give the smallest hyperharmonic integer $h_{n}^{(r)}$ greater than 1, which is obtained when $n = 33$ and $r = 10\, 667\, 968$.

Assume an almost huge cardinal with Mahlo target exists. We construct a model of ZFC in which a small cardinal carries a strongly centered filter by forcing with an iteration of two nested products of Levy collapses.

A system of nonlinear differential equations $x^{1+\gamma} (dY/dx) = F_{0}(x) + A(x)Y + F(x, Y)$ $(\gamma \geq 1)$ is considered. The origin $x = 0$ is irregular singular. There exist pioneering works about them. We study more precisely than preceding works, the meaning of asymptotic expansion of transformations and solutions by using Borel summable functions in asymptotic analysis and construct exponential series solutions often called transseries.

By a lot of previous work, it is known that the zeros of the period polynomial for a newform $f \in S_{k}(\Gamma_0(N))$ all lie on the circle $|z| = 1/\sqrt{N}$. In this paper we show that these zeros satisfy various interlacing properties for fixed $N$ and varying $k$ when either $k$ or $N$ is large. We also present a complete result when $N = 1$. Lastly, we establish the interlacing properties when $k$ is fixed and $N$ varies.

For virtual knot theory, the virtual braid group was defined by generalizing the braid group. It was proved that any virtual link can be obtained by the closure of a virtual braid. On the other hand, due to work by Jones et al., it is known that any (oriented) link is constructed from an element of Thompson's group $F$. In this paper, we define the “virtual version” of Thompson's group $F$ and prove that any virtual link is constructed from an element of the group.