Let $N \triangleleft G$ where $G$ is a finite group and let $B$ be a $p$-block of $G$, where $p$ is a prime. A Brauer character $\psi \in \mathop{\mathrm{IBr}}_{p}(B)$ is said to be of relative height zero with respect to $N$ provided that the height of $\psi$ is equal to that of an irreducible constituent of $\psi_{N}$. Now assume $G$ is $p$-solvable. In this paper, we count the number of relative height zero irreducible Brauer characters of $B$ with respect to $N$ that lie over any given $\varphi \in \mathop{\mathrm{IBr}}_{p}(N)$. As a consequence, we show that if $D$ is a defect group of $B$ and $\hat{B}$ is the unique $p$-block of $NN_{G}(D)$ with defect group $D$ such that $\hat{B}^{G} = B$, then $B$ and $\hat{B}$ have equal numbers of relative height zero irreducible Brauer characters with respect to $N$.

We study entire solutions for a discrete diffusive equation with bistable convolution type nonlinearity. We construct three different types of entire solutions. Each of these entire solutions behaves as two traveling wavefronts connecting two of those three equilibria as time approaches minus infinity. Moreover, the first and second ones are solutions which behave as two traveling wavefronts approaching each other from both sides of $x$-axis. The behavior of the second one is like the first one except it connects two different wavefronts. The third one is a solution which behaves as two different traveling wavefronts and one chases another from the same side of $x$-axis.

We give a classification of the translation hypersurfaces with constant mean curvature or constant Gauss--Kronecker curvature in Euclidean space or Lorentz--Minkowski space. We also characterize the minimal translation hypersurfaces in the upper half-space model of hyperbolic space.

We construct a connected, irreducible component of the moduli space of minimal surfaces of general type with $p_{g} = q = 2$ and $K^{2} = 5$, which contains both examples given by Chen--Hacon and the first author. This component is generically smooth of dimension $4$, and all its points parametrize surfaces whose Albanese map is a generically finite triple cover.

The indicator function of the set of $k$-th power free integers is naturally extended to a random variable $X^{(k)}({}\cdot{})$ on $(\hat{\mathbb{Z}},\lambda)$, where $\hat{\mathbb{Z}}$ is the ring of finite integral adeles and $\lambda$ is the Haar probability measure. In the previous paper, the first author noted the strong law of large numbers for $\{X^{(k)}({}\cdot{}+n)\}_{n=1}^{\infty}$, and showed the asymptotics: $E^{\lambda}[(Y_{N}^{(k)})^{2}] \asymp 1$ as $N \to \infty$, where $Y_{N}^{(k)}(x) := N^{-1/2k} \sum_{n=1}^{N} (X^{(k)}(x+n) - 1/\zeta(k))$. In the present paper, we prove the convergence of $E^{\lambda}[(Y_{N}^{(k)})^{2}]$. For this, we present a general proposition of analytic number theory, and give a proof to this.

In this paper, by modifying Cheng--Yau's technique to complete spacelike hypersurfaces in the de Sitter ($n+1$)-space $S_{1}^{n+1}(1)$, we prove a rigidity under the hypothesis of the mean curvature and the normalized scalar curvature being linearly related. As a corollary, we have the Theorem 1.1 of [3].

In this paper we study and obtain some necessary and sufficient conditions on the data for the existence, uniqueness of the strict solution and maximal regularity for some second-order differential equations with mixed boundary conditions whose forcing term belongs to Hölder continuous spaces. A few illustrative examples related to the interpolation theory are discussed.

Let $M$ be the three dimensional complete simply connected manifold of constant sectional curvature $0,1$ or $-1$. Let $\mathcal{L}$ be the manifold of all (unparametrized) complete oriented geodesics of $M$, endowed with its canonical pseudo-Riemannian metric of signature $(2,2)$ and Kähler structure $J$. A smooth curve in $\mathcal{L}$ determines a ruled surface in $M$. We characterize the ruled surfaces of $M$ associated with the magnetic geodesics of $\mathcal{L}$, that is, those curves $\sigma$ in $\mathcal{L}$ satisfying $\nabla_{\dot{\sigma}}\dot{\sigma}=J\dot{\sigma}$. More precisely: a time-like (space-like) magnetic geodesic determines the ruled surface in $M$ given by the binormal vector field along a helix with positive (negative) torsion. Null magnetic geodesics describe cones, cylinders or, in the hyperbolic case, also cones with vertices at infinity. This provides a relationship between the geometries of $\mathcal{L}$ and $M$.

We establish pointwise estimates for the ground states of some classes of positivity preserving operators. The considered operators are negatively perturbed (by measures) strongly local Dirichlet operators. These estimates will be written in terms of the Green's kernel of the considered operators, whose existence will be proved. In many circumstances our estimates are even sharp so that they recover known results about the subject. The results will deserve to obtain large time heat kernel estimates for the related operators.

We show that any $(\mathbb{C}^{*})^{n}$-invariant stably complex structure on a topological toric manifold of dimension $2n$ is integrable. We also show that such a manifold is weakly $(\mathbb{C}^{*})^{n}$-equivariantly isomorphic to a toric manifold.

The aim of this paper is to give families of Pisot and Salem elements $\beta$ in $\mathbb{F}_{q}((x^{-1}))$ with the curious property that the $\beta$-expansion of any rational series in the unit disk $D(0,1)$ is purely periodic. In contrast, the only known family of reals with the last property are quadratic Pisot numbers $\beta>1$ that satisfy $\beta^{2} = n\beta+1$ for some integer $n \geq 1$.

Akiyama et al. [2] proved an asymptotic formula for the distribution of CNS polynomials with fixed constant term. The objective of the present paper is to improve that result by providing an error term too.

We show that, as in de Rham cohomology over the complex numbers, the value of the entropy of an automorphism of the surface over a finite field $\mathbb{F}_{q}$ is taken on the span of the Néron--Severi group inside of $\ell$-adic cohomology.