In this paper we consider $O(m)\times O(n)$-invariant parabolic hypersurfaces of Euclidean space with constant mean curvature. We analyse the orbit space of the $O(m)\times O(n)$-group on $\mathbb{R}^{m+n}$ to give some classification results of these hypersurfaces for which the Gauss-Kronecker curvature does not change its sign.

Let $D$ be the open disk $|x|< R$ of a complete ultrametric algebraically closed field $\mathbb K$. We define the growth order $\rho(f)$, the growth type $\sigma(f)$ the cotype $\psi(f)$ and another expression $\theta(f)$ of an analytic function in $D$ and we show relations between the number of zeros of $f$ and the ``maximum modulus'' of $f$ involving $\rho,\sigma,\psi$. Then $\rho(f)$ lies in $[\theta(f)-1, \theta(f)]$. Moreover, if $0< \rho(f)< +\infty$ and $0< \psi(f)< +\infty$, then $\theta(f)=\rho(f)$ and $\sigma(f)=0$. Suppose $\mathbb K$ has characteristic zero and consider two unbounded analytic functions $f,g$ in $D$. The number of perfectly branched values admits overbounds linked to $\rho(f), \rho(g), \sigma(f), \sigma(g)$. In residue characteristic zero, then $\rho(f')=\rho(f),\sigma(f')=\sigma(f),$ $ \psi(f')=\psi(f)$, which can be applied to Levi-Civta fields.

We study a dynamic problem describing the frictional contact between a thermoviscoelastic body and a foundation. The thermoviscoelastic constitutive law includes a damage effect described by the parabolic inclusion with the homogeneous Neumann boundary condition and a temperature effect described by the first order evolution equation. The contact is modeled with normal compliance condition with friction. We present a variational formulation of the problem and establish an existence and uniqueness of the weak solution. The proof is based on parabolic variational inequalities of first and second kind, first order evolutionary variational equations and fixed point arguments.

In this paper, we study properties of asymptotic resemblance relations induced by compatible coarse structures on groups. We generalize the notion of asymptotic dimensiongrad for groups with compatible coarse structures and show this notion is coarse invariant. We end by defining the notion of set theoretic coupling for groups with compatible coarse structures and showing this notion is the generalization of the notion of topological coupling for finitely generated groups. We show if two groups with compatible coarse structures admit a set theoretic coupling then they are asymptotic equivalent.

The distance Laplacian spectral radius of a connected graph $G$ is the largest eigenvalue of its distance Laplacian matrix $\mathcal{L}(G)$ defined as $\mathcal{L}(G)=Tr(G)-D(G)$, where $Tr(G)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$. We determine the unique trees with maximum distance Laplacian spectral radius among trees of perfect matching with given maximum degree, the unique trees with second (third, respectively) maximum distance Laplacian spectral radius, and the unique bipartite unicyclic graphs with maximum distance Laplacian spectral radius. We also determine the unique graphs with minimum distance Laplacian spectral radius among bicyclic graphs.

Given a vector subspace of a topological algebra, along with a subset of its spectrum, we characterize the Choquet boundary of the restriction of the Gel'fand transform algebra on the previous subset, relative to the ``Gel'fand transform'' of the given subspace, restricted on the same subset. This is accomplished, by employing the geometric hull of the above subset and the respective strong points of the algebra, relative to the subspace involved. We also compare the relative Choquet boundaries for different subsets of the spectrum, via appropriate identifications, so that the closed convex hulls of the spectra, of the algebra and of its restricted Gel'fand transform, coincide. Finally, the Choquet boundaries of the latter algebras are also compared, in the case the above subset is a weakly peak set.

Let R be a commutative ring with identity. An $R$-module $M$ is called a semi-V-module if every nonzero homomorphic image of $M$ contains a nonzero V-submodule. An $R$-module $M$ is called a Bass module if every nonzero module in $\sigma [M]$ has a maximal submodule. It is shown that an $R$-module $M$ is a semi-V-module if and only if every nonzero cyclic submodule of $M$ is a Bass module if and only if $R/Ann_R(x)$ is a Bass ring for every nonzero element $x \in M$. Among other results, we characterize the class of rings $R$ for which every semi-V-module is a V-module and the class of rings $R$ for which every semi-V-module is a semisimple module. Also, we characterize the rings $R$ for which the class of semi-V-modules (Bass modules) is closed under direct products.

The concept of 2-nil ideal of a commutative ring $R$ is introduced, as an alternative to the $(2,n)$-ideals from [6]. We study its relationship with previously introduced classes of ideals, such as 2-absorbing ideals and $n$-ideals. Various properties and examples are presented. Our main result is a characterization of rings for which every 2-nil ideal is prime. We give a description of 2-nil ideals of general ZPI-rings.

A regular parallelism of real projective 3-space $\mathrm{PG}(3,\mathbb R)$ is an equivalence relation on the line space such that every class is equivalent to the set of 1-dimensional complex subspaces of $\mathbb C^2 = \mathbb R^4$. We shall assume that the set of classes is compact, and characterize those regular parallelisms that admit an action of a 2-dimensional torus group. We prove that there is a one-dimensional subtorus fixing every parallel class. From this property alone we deduce that the parallelism is a 2- or 3-dimensional regular parallelism in the sense of Betten and Riesinger [6]. If a 2-torus acts, then the parallelism can be described using a so-called generalized line star or \it gl star \rm which admits a 1-torus action. We also study examples of such parallelisms by constructing gl stars. In particular, we prove a claim which was presented in [6] with an incorrect proof. The present article continues a series of papers by the first author on parallelisms with large groups.