We study an initial boundary value problem on a ball for the heat-conductive system of compressible Navier–Stokes–Fourier equations, in particular, a criterion for the breakdown of the classical solution. For smooth initial data away from vacuum, we prove that the classical solution which is spherically symmetric loses its regularity in a finite time if and only if the density concentrates or vanishes or the velocity becomes unbounded around the center. One possible situation is that a vacuum ball appears around the center and the density may concentrate on the boundary of the vacuum ball simultaneously.

Lambrechts, Turchin, and Volić proved the Bousfield–Kan-type rational homology spectral sequence associated to the $d$th Kontsevich operad collapses at $E^2$-page if $d\ge 4$. The key of their proof is formality of the operad. In this paper, we simplify their proof by using a model category of operads. As by-products we obtain two new consequences. One is collapse of the spectral sequence in the case of $d=3$ (and the coefficients being rational numbers). The other says there is no nontrivial extension for the Gerstenhaber algebra structure on the spectral sequence.

We classify minimal algebraic surfaces of general type having $K^2=2\chi -1$ and $\chi \ge 7$. Such surfaces are regular with canonical map of degree one or two. If $p_g\ge 13$, then the surface is a genus-two fibration; otherwise we use the canonical map to describe these surfaces as birational either to the canonical image or to a double cover of a rational surface.

Studied are the composition series of the standard Whittaker $(\mathfrak{g},K)$-modules. For a generic infinitesimal character, the structures of these modules are completely understood, but if the infinitesimal character is integral, then there are not so many cases in which the structures of them are known. In this paper, as an example of the integral case, we determine the socle filtrations of the standard Whittaker $(\mathfrak{g},K)$-modules when $G$ is the group $Spin(r,1)$ and the infinitesimal character is regular integral.

In this paper, we prove the “local $\epsilon$-isomorphism” conjecture of Fukaya and Kato for a particular class of Galois modules, obtained by interpolating the twists of a fixed crystalline representation of $G_{{\mathbf{Q}}_p}$ by a family of characters of $G_{{\mathbf{Q}}_p}$. This can be regarded as a local analogue of the Iwasawa main conjecture for abelian $p$-adic Lie extensions of ${\mathbf{Q}}_p$, extending earlier work of Kato for rank one modules and of Benois and Berger for the cyclotomic extension. We show that such an $\epsilon$-isomorphism can be constructed using the 2-variable version of the Perrin-Riou regulator map constructed by the first and third authors.

Let $\Gamma$ be a finite group, and let $\Lambda$ be any Artin algebra. It is shown that the group algebra $\Lambda \Gamma$ is virtually Gorenstein if and only if $\Lambda \Gamma \text{'}$ is virtually Gorenstein, for all elementary abelian subgroups $\Gamma \text{'}$ of $\Gamma$. We also extend this result to cover the more general context. Precisely, assume that $\Gamma$ is a group in Kropholler’s hierarchy $\mathbf{H}\mathfrak{F}$, $\Gamma \text{'}$ is a subgroup of $\Gamma$ of finite index, and $R$ is any ring with identity. It is proved that, in certain circumstances, that $R\Gamma$ is virtually Gorenstein if and only if $R\Gamma \text{'}$ is so.

A $3$-dimensional generic flow is a pair $(M,v)$ with $M$ a smooth compact oriented $3$-manifold and $v$ a smooth nowhere-zero vector field on $M$ having generic behavior along $\partial M$; on the set of such pairs we consider the equivalence relation generated by topological equivalence (homeomorphism mapping oriented orbits to oriented orbits) and by homotopy with fixed configuration on the boundary, and we denote by $\mathcal{F}$ the quotient set. In this paper we provide a combinatorial presentation of $\mathcal{F}$. To do so we introduce a certain class $\mathcal{S}$ of finite $2$-dimensional polyhedra with extra combinatorial structures, and some moves on $\mathcal{S}$, exhibiting a surjection $\phi :\mathcal{S}\to \mathcal{F}$ such that $\phi \left(P_0\right)=\phi \left(P_1\right)$ if and only if $P_0$ and $P_1$ are related by the moves. To obtain this result we first consider the subset ${\mathcal{F}}_0$ of $\mathcal{F}$ consisting of flows having all orbits homeomorphic to closed segments or points, constructing a combinatorial counterpart ${\mathcal{S}}_0$ for ${\mathcal{F}}_0$, and then adapting it to $\mathcal{F}$.

Let $R$ be a commutative Noetherian ring, let $I$ be an ideal of $R$, and let $M$, $N$ be two finitely generated $R$-modules. The aim of this paper is to investigate the $I$-cofiniteness of generalized local cohomology modules $H_I^j(M,N)={\underset{⟶}{\lim }}_n{Ext}_R^j(M/I^{n}M,N)$ of $M$ and $N$ with respect to $I$. We first prove that if $I$ is a principal ideal, then $H_I^j(M,N)$ is $I$-cofinite for all $M$, $N$ and all $j$. Secondly, let $t$ be a nonnegative integer such that $dimSupp\left(H_I^j\right(M,N\left)\right)\le 1$ for all $j<t$. Then $H_I^j(M,N)$ is $I$-cofinite for all $j<t$ and $Hom(R/I,H_I^t(M,N\left)\right)$ is finitely generated. Finally, we show that if $dim\left(M\right)\le 2$ or $dim\left(N\right)\le 2$, then $H_I^j(M,N)$ is $I$-cofinite for all $j$.

Let $\mathbf{k}$ be a field. A field generator is a polynomial $F\in \mathbf{k}[X,Y]$ satisfying $\mathbf{k}(F,G)=\mathbf{k}(X,Y)$ for some $G\in \mathbf{k}(X,Y)$. If $G$ can be chosen in $\mathbf{k}[X,Y]$, we call $F$ a good field generator; otherwise, $F$ is a bad field generator. These notions were first studied by Abhyankar, Jan, and Russell in the 1970s. The present paper introduces and studies the notions of “very good” and “very bad” field generators. We give theoretical results as well as new examples of bad and very bad field generators.

Let $G={\mathbb{H}}_{2n+1}^r$ be the $(2n+1)$-dimensional reduced Heisenberg group, and let $H$ be an arbitrary connected Lie subgroup of $G$. Given any discontinuous subgroup $\Gamma \subset G$ for $G/H$, we show that resulting deformation space $\mathcal{T}(\Gamma ,G,H)$ of the natural action of $\Gamma$ on $G/H$ is endowed with a smooth manifold structure and is a disjoint union of open smooth manifolds. Unlike the setting of simply connected Heisenberg groups, we show that the stability property holds and that any discrete subgroup of $G$ is stable, following the notion of stability. On the other hand, a local (and hence global) rigidity theorem is obtained. That is, the related parameter space $\mathcal{R}(\Gamma ,G,H)$ admits a rigid point if and only if $\Gamma$ is finite.