Let $G$ be a finite group, $H$ a normal subgroup of $G$ and $b$ and $c$ block idempotents of $\mathcal{O}G$ and $\mathcal{O}H$ respectively. Under the assumption that $C_{H}(R)\subset O_{p',p}(H)$ for a Sylow $p$-subgroup $R$ of $O_{p',p}(H)$ and $c$ is also a block idempotent of $\mathcal{O}O_{p'}(H)$, we give two equivalent conditions about when $\mathcal{O}Gb$ and $\mathcal{O}Hc$ are natural Morita equivalent of degree $n$ (see Theorem 1.5).

We study the behavior of generic initial ideals with respect to fibre products. In our main result we determine the generic initial ideal of the fibre product with respect to the reverse lexicographic order. As an application we compute explicitly the generic initial ideal of a fibre product in a special case. We also prove that the fibre product of two graded ideals is componentwise linear if and only if both ideals have this property.

We study a nonlocal parabolic problem modeling the temperature in a thin region during linear friction welding for a hard material. We derive the structures of steady states of this nonlocal problem and its associated approximated problems. Moreover, some remarks on the parabolic problem are given.

In this paper we study a class of stochastic partial differential equations in the whole space $\mathbb{R}^{d}$, with arbitrary dimension $d\geq 1$, driven by a Gaussian noise white in time and correlated in space. The differential operator is a fractional derivative operator. We show the existence, uniqueness and Hölder's regularity of the solution. Then by means of Malliavin calculus, we prove that the law of the solution has a smooth density with respect to the Lebesgue measure.

We show that any lexsegment ideal with linear resolution has linear quotients with respect to a suitable ordering of its minimal monomial generators. For completely lexsegment ideals with linear resolution we show that the decomposition function is regular. For arbitrary lexsegment ideals we compute the depth and the dimension. As application we characterize the Cohen--Macaulay lexsegment ideals.

In this paper we prove the existence, uniqueness and stability of solutions of an inverse problem to a time-dependent Ginzburg--Landau model for superconductivity with the final overdetermination.

A quasitoric manifold (resp. a small cover) is a $2n$-dimensional (resp. an $n$-dimensional) smooth closed manifold with an effective locally standard action of $(S^{1})^{n}$ (resp. $(\mathbb{Z}_{2})^{n}$) whose orbit space is combinatorially an $n$-dimensional simple convex polytope $P$. In this paper we study them when $P$ is a product of simplices. A generalized Bott tower over $\mathbb{F}$, where $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$, is a sequence of projective bundles of the Whitney sum of $\mathbb{F}$-line bundles starting with a point. Each stage of the tower over $\mathbb{F}$, which we call a generalized Bott manifold, provides an example of quasitoric manifolds (when $\mathbb{F}=\mathbb{C}$) and small covers (when $\mathbb{F}=\mathbb{R}$) over a product of simplices. It turns out that every small cover over a product of simplices is equivalent (in the sense of Davis and Januszkiewicz [5]) to a generalized Bott manifold. But this is not the case for quasitoric manifolds and we show that a quasitoric manifold over a product of simplices is equivalent to a generalized Bott manifold if and only if it admits an almost complex structure left invariant under the action. Finally, we show that a quasitoric manifold $M$ over a product of simplices is homeomorphic to a generalized Bott manifold if $M$ has the same cohomology ring as a product of complex projective spaces with $\mathbb{Q}$ coefficients.

Each closed oriented 3-manifold $M$ is naturally associated with a set of integers $D(M)$, the degrees of all self-maps on $M$. $D(M)$ is determined for each torus bundle and semi-bundle $M$. The structure of torus semi-bundle is studied in detail. The paper is a part of a project to determine $D(M)$ for all 3-manifolds in Thurston's picture.

We consider a class of quasilinear fuchsian operators $Q$ of order $m\geq 1$, holomorphic in a neighborhood of the origin in $\mathbf{C}_{t} \times \mathbf{C}_{x}^{n}$, and having a simple characteristic hypersurface transverse to $S$: $t=0$. Under an assumption on the linear part of $Q$, we construct solutions of the problem $Qu=v$ in spaces of ramified functions of slow growth. The result is an extension of [15] to the quasilinear case.

We show that an upper bound for the maximal Thurston--Bennequin number of any double of a knot $K$ given by the Kauffman polynomial is sharp if the bound is sharp for $K$. In particular, we give formulas for the maximal Thurston--Bennequin numbers of positive doubles of torus knots and two-bridge knots.

Gordon and Luecke showed that knots are determined by their complements. Therefore a non-trivial Dehn surgery on a non-trivial knot does not yield the 3-sphere. But the situation for links is different from that for knots. Berge constructed some examples of Dehn surgeries of 2-component links yielding the 3-sphere with interesting properties. By extending Berge's example, we construct infinitely many examples of tunnel number one links in the 3-sphere, such that their components are non-trivial, and that non-trivial Dehn surgeries on them yield the 3-sphere.

A recollement describes one triangulated category $\mathrm{\mathsf{T}}$ as ``glued together'' from two others, $\mathrm{\mathsf{S}}$ and $\mathrm{\mathsf{U}}$. The definition is not symmetrical in $\mathrm{\mathsf{S}}$ and $\mathrm{\mathsf{U}}$, but this note shows how $\mathrm{\mathsf{S}}$ and $\mathrm{\mathsf{U}}$ can be interchanged when $\mathrm{\mathsf{T}}$ has a Serre functor.

In 1960, P.A. Smith raised an isomorphism problem. Is it true that the tangential $G$-modules at two fixed points of an arbitrary smooth $G$-action on a sphere with exactly two fixed points are isomorphic to each other? Given a finite group, the Smith set of the group means the subset of real representation ring consisting of all differences of Smith equivalent representations. Many researchers have studied the Smith equivalence for various finite groups. But the Smith sets for non-perfect groups were rarely determined. In particular, the Smith set for a non-gap group has not been determined unless it is trivial. We determine the Smith set for the non-gap group $G = S_{5} \times C_{2} \times \dots \times C_{2}$.

This work is devoted to the Marsden--Weinstein reduction problem for Kaehlerian metrics [34]. We focus the attention on Kaehlerian manifolds $(M,\omega,J)$ whose underlying symplectic structures $(M,\omega)$ are homogeneous under symplectic actions of completely solvable Lie groups. Homogeneous symplectic manifolds of this type always admit lagrangian foliations. In this setting the starting Kaehlerian reduction problem yields new problems involving the topology of affinely flat manifolds [36]. The KV cohomology of these affinely flat structures is computed using techniques of spectral sequences. Questions close to bi-lagrangian structures are discussed. The KV cohomology of affinely flat manifolds is used to give new proofs of some known deep theorems. For instance we revisit the fundamental conjecture of Gindikin, Piateccii--Sapiro and Vinberg [18], [19]. The global geometry of compact symplectic solvmanifolds admitting Kaehlerian metrics is revisited. Various additonal items which are closely related to the reduction problem and to lagrangian foliations are discussed.

Ce travail est consacré à l'analogue Kählérien du problème de réduction des variétés symplectiques d'après Marsden-Weinstein [34]. L'accent est mis sur les variétés Kählériennes $(M,\omega,J)$ dont les structures symplectiques sous-jacentes $(M,\omega)$ sont homogènes sous des actions symplectiques des groupes de Lie complètement résolubles. Les variétés symplectiques homogènes de ce type portent toujours des feuilletages lagrangiens. Du problème initial de réduction Kählérienne surgisent des nouvelles questions liées à la topologie des variétés affinement plates [36]. La KV cohomologie de ces structures affinement plates est calculée par des techniques de suite spectrale. Des questions en relation avec les structures bi-lagangiennes sont discutées. Dans ce même contexte la KV cohomologie est utilisée pour donner des nouvelles démonstrations de certaines conjectures devenues théorèmes. C'est le cas de la conjecture fondamentale de Gindikin, Piateccii-Sapiro et Vinberg [18], [19]. Nous avons aussi revisité la géométrie globale des soluvariétés qui admettent des métriques Kählériennes. D'autres prolongements sont explorés mais non exploités.

This is the first of a series of papers which will be devoted to the study of the extended $G$-actions on torus manifolds $(M^{2n}, T^{n})$, where $G$ is a compact, connected Lie group whose maximal torus is $T^{n}$. The goal of this paper is to characterize codimension $0$ extended $G$-actions up to essential isomorphism. For technical reasons, we do not assume that torus manifolds are omnioriented. The main result of this paper is as follows: a homogeneous torus manifold $M^{2n}$ is (weak equivariantly) diffeomorphic to a product of complex projective spaces $\prod\mathbb{C}P(l)$ and quotient spaces of a product of spheres $\bigl(\prod S^{2m}\bigr)/\mathcal{A}$ with standard torus actions, where $\mathcal{A}$ is a subgroup of $\prod \mathbb{Z}_{2}$ generated by the antipodal involutions on $S^{2m}$. In particular, if the homogeneous torus manifold $M^{2n}$ is a compact (non-singular) toric variety or a quasitoric manifold, then $M^{2n}$ is just a product of complex projective spaces $\prod \mathbb{C}P(l)$.

In this paper, we show that the orbit space of the domain by the elliptic Weyl group for an elliptic root system of codimension 1 has a structure of Frobenius manifold. We also give a characterization of this structure of the Frobenius manifold under suitable conditions.