Let $\{u_{t}(x)\colon t\geq 0{,}\ x\in \mathbb{R}\}$ be a random string taking values in $\mathbb{R}^{d}$. It is specified by the following stochastic partial differential equation, \begin{equation*} \frac{\partial u_{t}(x)}{\partial t} =\frac{\partial^{2}u_{t}(x)}{\partial x^{2}}+\dot{W}, \end{equation*} where $\dot{W}(x,t)$ is two-parameter white noise. The objective of the present paper is to study the fractal properties of the algebraic sum of the image sets for the random string process $\{u_{t}(x)\colon t\geq 0{,}\ x\in \mathbb{R}\}$. We obtain the Hausdorff and packing dimensions of the algebraic sum of the image sets of the string. We also consider the existence of the local times of the process $\{u_{s}(y)-u_{t}(x)\colon s,t\geq 0\colon x, y\in \mathbb{R}\}$, and find the Hausdorff and packing dimensions of the level sets for the process $\{u_{s}(y)-u_{t}(x)\colon s,t\geq 0; x, y\in \mathbb{R}\}$.

We prove that a nilpotent block of a finite group and its Glauberman correspondent are basic Morita equivalent.

Consider a Riemannian vector bundle of rank 1 defined by a normal vector field $\nu$ on a surface $M$ in $\mathbb{R}^{4}$. Let $\mathrm{II}_{\nu}$ be the second fundamental form with respect to $\nu$ which determines a configuration of lines of curvature. In this article, we obtain conditions on $\nu$ to isometrically immerse the surface $M$ with $\mathrm{II}_{\nu}$ as a second fundamental form into $\mathbb{R}^{3}$. Geometric restrictions on $M$ are determined by these conditions. As a consequence, we analyze the extension of Loewner's conjecture, on the index of umbilic points of surfaces in $\mathbb{R}^{3}$, to special configurations on surfaces in $\mathbb{R}^{4}$.

In this note we study finite semifield planes which admit an irreducible planar Baer collineation. This continues previous work of N. Johnson [5].

In this paper, we consider the Cauchy problem \begin{equation*} \left\{ \begin{array}{@{}ll@{}} u_{t}=(u^{m-1}u_{x})_{x}, & x \in \mathbb{R}{,}\ t>0{,}\ {-1} < m \leq 1,\\ u(x,0)=u_{0}, & x \in \mathbb{R}. \end{array} \right. \end{equation*} We will prove that: \begin{enumerate}[1)] \item $|u(x,t,m)-u(x,t,m_{0})|\to 0$ uniformly on $[-l,l]\times[\tau,T]$ as $m \to m_{0}$ for any given $l > 0$, $0<\tau<T$ and $-1<m$, $m_{0}<1$, \item $\int_{\mathbb{R}}|u(x,t,m)-u(x,t,1)|\,dx \leq 2((1-m)/m)\|u_{0}\|_{L^{1}(\mathbb{R})}$. \end{enumerate}

We study the wellposedness in the Gevrey classes $\gamma^{s}$ and in $C^{\infty}$ of the Cauchy problem for 2 by 2 weakly hyperbolic systems. In this paper we shall give some conditions to the case that the characteristic roots oscillate rapidly and vanish at an infinite number of points.

The notion of negative slope algorithm was introduced by S. Ferenczi, C. Holton, and L. Zamboni in their study of the three letter languages arising from three interval exchange transformations. They characterize eventually periodic points of the negative slope algorithm. In this paper, by using the natural extension of the negative slope algorithm, we give a necessary and sufficient condition for purely periodicity of the negative slope algorithm.

We use results on the local theta correspondence to prove that for large degrees the Duke-Imamoglu-Ikeda lifting of an elliptic modular form is not a linear combination of theta series.

We study the structure of symplectic quandles, quandles which are also $R$-modules equipped with an antisymmetric bilinear form. We show that every finite dimensional symplectic quandle over a finite field $\mathbb{F}$ or arbitrary field $\mathbb{F}$ of characteristic other than 2 is a disjoint union of a trivial quandle and a connected quandle. We use the module structure of a symplectic quandle over a finite ring to refine and strengthen the quandle counting invariant.

Asymptotic behavior of solutions to the compressible Navier-Stokes equation around a given constant state is investigated on a cylindrical domain in $\mathbf{R}^{3}$, under the no slip boundary condition for the velocity field. The $L^{2}$ decay estimate is established for the perturbation from the constant state. It is also shown that the time-asymptotic leading part of the perturbation is given by a function satisfying a 1 dimensional heat equation. The proof is based on an energy method and asymptotic analysis for the associated linearized semigroup.

In this paper we present a generalisation of a theorem due to Cornalba and Harris, which is an application of Geometric Invariant Theory to the study of invariants of fibrations. In particular, our generalisation makes it possible to treat the problem of bounding the invariants of general fibred surfaces. As a first application, we give a new proof of the slope inequality and of a bound for the invariants associated to double cover fibrations.

We formalize various properties of characteristic functors on $p$-groups, and discuss relationships between them. Applications to the Thompson subgroup and certain of its analogues are then given.

We study theory of curves in the complex hyperbola and show that special motions of curves are linked with the Burgers hierarchy, which also leads to a Hamiltonian formulation of the hierarchy and to the difference Burgers equation via their discretization.

In this paper we show that a uniruled manifold with a split tangent bundle admits almost holomorphic fibrations that are related to the splitting. We analyse these fibrations in detail in several special cases. This yields new results about the integrability of the direct factors and the universal covering of the manifold.

We prove the colored hook formula for a finite pre-dominant integral weight. As a corollary of this, we get a new proof of the Peterson's hook formula.