In this paper we consider a free boundary problem for spacelike surfaces in the 3-dimensional Lorentz-Minkowski space $\mathbb{L}^{3}$ whose energy functional involves the area of a surface and a timelike potential. The critical points of this energy for any volume-preserving admissible variation are spacelike surfaces supported in a plane and whose mean curvature is a linear function of the time coordinate. In this paper, we consider those surfaces that are invariant in a parallel coordinate to the support plane. We call these surfaces stationary bands. We establish existence of such surfaces and we investigate their qualitative properties. Finally, we give estimates of its size in terms of the initial data.

Let $G$ be a compact connected Lie group and $M$ a rational cohomology complex quadric of real dimension divisible by $4$ (where $\dim M\neq 4$). The aim of this paper is to classify pairs $(G,M)$ such that $G$ acts smoothly on $M$ with codimension one principal orbits. There exist eight such pairs up to essential isomorphism. The underlying manifold $M$ is diffeomorphic to the genuine complex quadric except one pair.

In this paper, we introduce the weak $G$-expansivity which is a generalization of both expansivity and $G$-expansivity. Also, we define $G$-stable and $G$-unstable sets of a homeomorphism on a metric $G$-space $X$ and investigate properties of them. Finally, we consider the decomposition theorem on $G$-spaces.

In this note we will obtain defining equations of modular curves $X_{0}(2^{2n})$. The key ingredient is a recursive formula for certain generators of the function fields on $X_{0}(2^{2n})$.

Recently, Donaldson proved asymptotic stability for a polarized algebraic manifold $M$ with polarization class admitting a Kähler metric of constant scalar curvature, essentially when the linear algebraic part $H$ of $\operatorname{Aut}^{0}(M)$ is semisimple. The purpose of this paper is to give a generalization of Donaldson's result to the case where the polarization class admits an extremal Kähler metric, even when $H$ is not semisimple.

The article studies approximations for stable like jump processes on fractal sets $F\subset\mathbb{R}^{n}$. Processes on $d$-sets are approximated by jump processes on the $\varepsilon$-parallel sets. For the special case of self-similar sets with equal contraction ratios, approximations in terms of finite Markov chains are provided. In either case, the convergence of Dirichlet forms, semigroups and resolvents are established as well as the convergence of the finite-dimensional distributions under canonical initial distributions. In the self-similar case also the weak convergence of the laws under these initial distributions in $D_{F}([0,t_{0}])$ is proved.

We find the minimal number of non-trivial links in an embedding of any complete $k$-partite graph on 7 vertices (including $K_{7}$, which has at least 21 non-trivial links). We give either exact values or upper and lower bounds for the minimal number of non-trivial links for all complete $k$-partite graphs on 8 vertices. We also look at larger complete bipartite graphs, and state a conjecture relating minimal linking embeddings with minimal book embeddings.

For a knot $K$ in $\mathbb{S}^{3}$, Kakimizu introduced a simplicial complex whose vertices are all the isotopy classes of minimal genus spanning surfaces for $K$. The first purpose of this paper is to prove the $1$-skeleton of this complex has diameter bounded by a function quadratic in knot genus, whenever $K$ is atoroidal. The second purpose of this paper is to prove the intersection number of two minimal genus spanning surfaces for $K$ is also bounded by a function quadratic in knot genus, whenever $K$ is atoroidal. As one application, we prove the simple connectivity of Kakimizu's complex among all atoroidal genus $1$ knots.

In this paper we present sufficient conditions and necessary conditions for a single element to form a $p$-basis of a ring of constants of a derivation. We consider some special cases, when these conditions are equivalent, and we analyze some counter-examples, when the equivalence does not hold.

We compute the eigenvalues with multiplicities of the Lichnerowicz Laplacian acting on the space of symmetric covariant tensor fields on the Euclidian sphere $S^{n}$. The spaces of symmetric eigentensors are explicitly given.

Our aim in this paper is to give Sobolev's inequality and Trudinger exponential integrability for Riesz potentials of functions in non-doubling Morrey spaces.

We deal with the value distribution problem for the linear combinations of multiplicities of the cycle lengths of a random permutation. To examine the characteristic functions, we derive asymptotic formulas for ratios of the Taylor coefficients of the relevant generating series. The proposed version of analytic method does not require any analytic continuation of these series outside the convergence disk.

We study the boundary $L_{t}$ of the Milnor fiber for the non-isolated singularities in $\mathbf{C}^{3}$ with equation $z^{m} - g(x,y) = 0$ where $m \geq 2$ and $g(x,y)=0$ is a non-reduced plane curve germ. We give a complete proof that $L_{t}$ is a Waldhausen graph manifold and we provide the tools to construct its plumbing graph. As an example, we give the plumbing graph associated to the germs $z^{2} - (x^{2} - y^{3})y^{l} = 0$ with $l$ odd and $l \geq 3$. We prove that the boundary of the Milnor fiber is a Waldhausen manifold new in complex geometry, as it cannot be the boundary of a normal surface singularity.