The paper is concerned with linear thermoelastic plate equations in the half-space ${\mathit{R}}_{+}^n=\{x=(x_1,\dots ,x_n)❘x_n>0\}$:

$u_{\mathit{tt}}+{\mathrm{\Delta }}^{2}u+\mathrm{\Delta }\theta =0$ and

${\theta }_t-\mathrm{\Delta }\theta -\mathrm{\Delta }{u}_t=0$ ${\mathit{R}}_{+}^n\times (0,\infty ),$

subject to the boundary condition: $u{|}_{x_n=0}=D_{n}u{|}_{x_n=0}=\theta {|}_{x_n=0}=0$ and initial condition: $(u,D_{t}u,\theta ){|}_{t=0}=(u_0,v_0,{\theta }_0)\in {\mathcal{H}}_p=W_{p,D}^2\times L_p\times L_p$, where $W_{p,D}^2=\{u\in W_p^2❘u{|}_{x_n=0}=D_{n}u{|}_{x_n=0}=0\}$. We show that for any $p\in (1,\infty )$, the associated semigroup $\{T(t){\}}_{t\ge 0}$ is analytic in the underlying space ${\mathcal{H}}_p$. Moreover, a solution $(u,\theta )$ satisfies the estimates:

$\parallel {\nabla }^j({\nabla }^{2}u(·,t),u_t(·,t),\theta (·,t)){\parallel }_{L_q({\mathit{R}}_{+}^n)}$

$\le C_{p,q}{t}^{-\frac{j}{2}-\frac{n}{2}(\frac{1}{p}-\frac{1}{q})}\parallel ({\nabla }^2{u}_0,v_0,{\theta }_0){\parallel }_{L_p\left({\mathit{R}}_{+}^n\right)}$

$(t>0)$

for $j=0,1,2$ provided that when $j=0$, 1 and that when $j=2$, where ${\nabla }^j$ stands for space gradient of order $j$.

A classical problem in finite group theory dating back to Jordan, Klein, E. H. Moore, Dickson, Blichfeldt etc. is to determine all finite subgroups in $\mathit{SL}(n,\mathit{C})$ up to conjugation for some small values of $n$. This question is important in group theory as well as in the study of quotient singularities. Some results of Blichfeldt when $n=3,4$ were generalized to the case of finite primitive subgroups of $\mathit{SL}(5,\mathit{C})$ and $\mathit{SL}(7,\mathit{C})$ by Brauer and Wales. The purpose of this article is to consider the following case. Let $p$ be any odd prime number and $G$ be a finite primitive subgroup of $\mathit{SL}(p,\mathit{C})$ containing a non-trivial monomial normal subgroup $H$ so that $H$ has a non-scalar diagonal matrix. We will classify all these groups $G$ up to conjugation in $\mathit{SL}(p,\mathit{C})$ by exhibiting the generators of $G$ and representing $G$ as some group extensions. In particular, see the Appendix for a list of these subgroups when $p=5$ or 7.

We describe an approach to classification of weighted homogeneous Saito free divisors in ${\mathit{C}}^3$. This approach is mainly based on properties of Lie algebras of vector fields tangent to reduced hypersurfaces at their non-singular points. In fact we also obtain a classification of such Lie algebras having similar properties as ones for discriminants associated with irreducible real reflection groups of rank 3. Among other things we briefly discuss some applications to the theory of discriminants of irreducible reflection groups of rank 3, some interesting relationships with root systems of types $E_6$, $E_7$, $E_8$, and few examples in higher dimensional cases.

We prove the existence of extension dimension for a much expanded class of spaces. First we obtain several theorems which state conditions on a polyhedron or $\mathrm{CW}$-complex $K$ and a space $X$ in order that $X$ be an absolute co-extensor for $K$. Then we prove the existence of and describe a wedge representative of extension dimension for spaces in a wide class relative to polyhedra or $\mathrm{CW}$-complexes. We also obtain a result on the existence of a “countable” representative of the extension dimension of a Hausdorff compactum.

Let $V$ be a total valuation ring of a division ring $K$ with an automorphism $\sigma$ and let $A={\oplus }_{i\in \mathit{Z}}{A}_i{X}^i$ be a graded extension of $V$ in $K[X,X^{-1};\sigma ]$, the skew Laurent polynomial ring. We classify $A$ by distinguishing three different types based on the properties of $A_1$ and $A_{-1}$, and a complete description of $A_i$ for all $i\in \mathit{Z}$ is given in the case where $A_1$ is not a finitely generated left $O_l(A_1)$-ideal.

We compute the fundamental group of the complement of each irreducible sextic of weight eight or nine (in a sense, the largest groups for irreducible sextics), as well as of 169 of their derivatives (both of and not of torus type). We also give a detailed geometric description of sextics of weight eight and nine and of their moduli spaces and compute their Alexander modules; the latter are shown to be free over an appropriate ring.

In this paper, we consider generalized moment maps for Hamiltonian actions on $H$-twisted generalized complex manifolds introduced by Lin and Tolman [15]. The main purpose of this paper is to show convexity and connectedness properties for generalized moment maps. We study Hamiltonian torus actions on compact $H$-twisted generalized complex manifolds and prove that all components of the generalized moment map are Bott-Morse functions. Based on this, we shall show that the generalized moment maps have a convex image and connected fibers. Furthermore, by applying the arguments of Lerman, Meinrenken, Tolman, and Woodward [13] we extend our results to the case of Hamiltonian actions of general compact Lie groups on $H$-twisted generalized complex orbifolds.

We compute the generalized Lefschetz number of orientation-preserving self-homeomorphisms of a compact punctured disk, using the fact that homotopy classes of these homeomorphisms can be identified with braids. This result is applied to study Nielsen-Thurston canonical homeomorphisms on a punctured disk. We determine, for a certain class of braids, the rotation number of the corresponding canonical homeomorphisms on the outer boundary circle. As a consequence of this result on the rotation number, it is shown that the canonical homeomorphisms corresponding to some braids are pseudo-Anosov with associated foliations having no interior singularities.

In this note, we explicitly construct the twistor spaces of some Joyce metrics on the connected sum of arbitrary number of complex projective planes. Unlike our former construction for the case of four complex projective planes, the present construction mainly utilizes minitwistor spaces, and partially follows the method and construction given in [5] and [6].

Consider a geometrically finite Kleinian group $G$ without parabolic or elliptic elements, with its Kleinian manifold $M=({\mathit{H}}^3\bigcup {\Omega }_G)/G$. Suppose that for each boundary component of $M$, either a maximal and connected measured lamination in the Masur domain or a marked conformal structure is given. In this setting, we shall prove that there is an algebraic limit $\mathrm{\Gamma }$ of quasi-conformal deformations of $G$ such that there is a homeomorphism $h$ from $\mathrm{Int}M$ to ${\mathit{H}}^3/\mathrm{\Gamma }$ compatible with the natural isomorphism from $G$ to $\mathrm{\Gamma }$, the given laminations are unrealisable in ${\mathit{H}}^3/\mathrm{\Gamma }$, and the given conformal structures are pushed forward by $h$ to those of ${\mathit{H}}^3/\mathrm{\Gamma }$. Based on this theorem and its proof, in the subsequent paper, the Bers-Thurston conjecture, saying that every finitely generated Kleinian group is an algebraic limit of quasi-conformal deformations of minimally parabolic geometrically finite group, is proved using recent solutions of Marden’s conjecture by Agol, Calegari-Gabai, and the ending lamination conjecture by Minsky collaborating with Brock, Canary and Masur.

Generalizing the Ohio completeness property, we introduce the notion of $\kappa$-Ohio completeness. Although many results from a previous paper by the authors may easily be adapted for this new property, there are also some interesting differences. We provide several examples to illustrate this. We also have a consistency result; depending on the value of the cardinal $\mathfrak{d}$, the countable union of open and ${\omega }_1$-Ohio complete subspaces may or may not be ${\omega }_1$-Ohio complete.