Existence of nonnegative weak solutions is shown for a thin film approximation of the Muskat problem with gravity and capillary forces taken into account. The model describes the space-time evolution of the heights of the two fluid layers and is a fully coupled system of two fourth order degenerate parabolic equations. The existence proof relies on the fact that this system can be viewed as a gradient flow for the 2-Wasserstein distance in the space of probability measures with finite second moment.

We prove that a log del Pezzo surface of Picard number one contains at most $3$ singular points if its smooth locus is simply connected. This establishes the algebraic Montgomery-Yang problem for log del Pezzo surfaces.

We shall consider the maximal cyclotomic extension of a totally real finite algebraic number field and its ideal class group. We shall investigate the structure of the ideal class group with the action of the cyclotomic Galois group.

Let 0 < $\alpha$ < 1 be a transcendental real number and $\lambda_1,\ldots,\lambda_r$ be real numbers with $0 \le \lambda_j$ < 1. It is conjectured that a joint universality theorem for a collection of Lerch zeta functions $\{L(\lambda_j,\alpha,s)\}$ will hold for every numbers $\lambda_j$'s which are different each other. In this paper we will prove that the joint universality theorem for the set $\{L(\lambda_j,\alpha,s)\}$ holds for almost all real numbers $\lambda_j$'s.

We extend the $\lambda$-Aluthge sequence convergence theorem of Antezana, Pujals and Stojanoff in the context of real noncompact connected semisimple Lie groups.

In this article we deal with a class of degenerate parabolic systems that encompasses two different effects: porous medium and chemotaxis. Such classes of equations arise in the mesoscale level modeling of biomass spreading mechanisms via chemotaxis. We prove estimates related to the existence of the global attractor under certain `balance conditions' on the order of the porous medium degeneracy and the growth of the chemotactic function.

For an infinite cardinal $\tau$, let $\ell_2^f(\tau)$ be the linear span of the canonical orthonormal basis of the Hilbert space $\ell_2(\tau)$ of weight $= \tau$. In this paper, we give characterizations of topological manifolds modeled on $\ell_2^f(\tau)$ and $\ell_2^f(\tau) \times \bm{Q}$, where $\bm{Q} = [-1,1]^{\mathbb{N}}$ is the Hilbert cube. We denote the full simplicial complex of cardinality $= \tau$ and the hedgehog of weight $= \tau$ by $\Delta(\tau)$ and $J(\tau)$, respectively. Using our characterization of $\ell_2^f(\tau)$, we prove that both the metric polyhedron of $\Delta(\tau)$ and the space

$J(\tau)^{\mathbb{N}}_f = \{x \in J(\tau)^{\mathbb{N}} \mid x(n) = 0 \text{ except for finitely many } n \in \mathbb{N}\}$

are homeomorphic to $\ell_2^f(\tau)$.

Let $S$ be a finite group with a character, $\rm sgn$, of order 2, and $S'$ its central extension by a group $Z=\langle z\rangle$ of order 2. A representation $\pi$ of $S'$ is called {\it spin} if $\pi(z\sigma')=-\pi(\sigma')$ $(\sigma'\in S')$, and the set of all equivalence classes of spin irreducible representations (= IRs) of $S'$ is called the {\it spin dual} of $S'$. Take a finite number of such triplets $(S'_j,z_j,{\rm sgn}_j)$ $(1\le j\le m)$. We define twisted central product $S'=S'_1\hat{*}S'_2\hat{*}\cdots\hat{*}S'_m$ as a double covering of $S=S_1\times\cdots \times S_m$, $S_j=S'_j/\langle z_j\rangle$, and for spin IRs $\pi_j$ of $S'_j$, define twisted central product $\pi=\pi_1\hat{*}\pi_2\hat{*}\cdots\hat{*}\pi_m$ as a spin IR of $S'$. We study their characters and prove that the set of spin IRs $\pi$ of this type gives a complete set of representatives of the spin dual of $S'$. These results are applied to the case of representation groups $S'$ for $S={\mathfrak S}_n$ and ${\mathfrak A}_n$, and their (Frobenius-)Young type subgroups.

In this paper we deduce a local deformation lemma for uniform embeddings in a metric covering space over a compact manifold from the deformation lemma for embeddings of a compact subspace in a manifold. This implies the local contractibility of the group of uniform homeomorphisms of such a metric covering space under the uniform topology. Furthermore, combining with similarity transformations, this enables us to induce a global deformation property of groups of uniform homeomorphisms of metric spaces with Euclidean ends. In particular, we show that the identity component of the group of uniform homeomorphisms of the standard Euclidean $n$-space is contractible.

Let $p$ be a prime not equal to 2 or 3. In this paper we study the $\mathbb{Q}$-rational cuspidal group $\mathcal{C}_{\mathbb{Q}}$ of the jacobian $J_{1}(2p)$ of the modular curve $X_{1}(2p)$. We prove that the group $\mathcal{C}_{\mathbb{Q}}$ is generated by the $\mathbb{Q}$-rational cusps. We determine the order of $\mathcal{C}_{\mathbb{Q}}$, and give numerical tables for all $p\leq127$. These tables give also other cuspidal class numbers for the modular curves $X_{1}(2p)$ and $X_{1}(p)$. We give a basis of the group of the principal divisors supported on the $\mathbb{Q}$-rational cusps, and using this we determine the explicit structure of $\mathcal{C}_{\mathbb{Q}}$ for all $p\leq127$. We determine the structure of the Sylow $p$-subgroup of $\mathcal{C}_{\mathbb{Q}}$, and the explicit structure for all $p\leq4001$.

We study bridge number and tunnel number of tangles and knots, and also study their behavior under tangle decomposition of knots.

First, we consider some fundamental properties including dual spaces, complex interpolations of $\alpha$-modulation spaces $M^{s,\alpha}_{p,q}$ with 0 < $p$, $q \le \infty$. Next, necessary and sufficient conditions for the scaling property and the inclusions between $\alpha_1$-modulation and $\alpha_2$-modulation spaces are obtained. Finally, we give some criteria for $\alpha$-modulation spaces constituting multiplication algebra. As a by-product, we show that there exists an $\alpha$-modulation space which is not an interpolation space between modulation and Besov spaces. In a subsequent paper, we will give some applications of $\alpha$-modulation spaces to nonlinear dispersive wave equations.

In this article we develop a new approach to the problem of the stability of locally conformally Kähler structures (l.c.k structures) under small deformations of complex structures and deformations of flat line bundles. We show a cohomological criterion for the stability of l.c.k structures. We apply our approach to generalizations of Hopf manifolds to obtain the stability of l.c.k structures which do not have potential in general. We give an explicit description of the cohomological obstructions of the stability of l.c.k structures on Inoue surfaces with $b_2=0$.