Volume 26, Issue 3

Abstract. The Euler–Poisson system is a fundamental two-fluid model to describe the dynamics of the plasma consisting of compressible electrons and a uniform ion background. By using the dispersive Klein–Gordon effect, Guo (1998) first constructed a global smooth irrotational solution in the three-dimensional case. It has been conjectured that same results should hold in the two-dimensional case. The main difficulty in 2D comes from the slow dispersion of the linear flow and certain nonlocal resonant obstructions in the nonlinearity. In this paper we develop a new method to overcome these difficulties and construct smooth global solutions for the 2D Euler–Poisson system.

Abstract. The aim of this paper is to study the representation theory of quantum Schubert cells. Let 𝔤$\mathfrak {g}$ be a simple complex Lie algebra. To each element w of the Weyl group W of 𝔤$\mathfrak {g}$, De Concini, Kac and Procesi have attached a subalgebra Uq[w]$U_q[w]$ of the quantised enveloping algebra Uq(𝔤)$U_q(\mathfrak {g})$. Recently, Yakimov showed that these algebras can be interpreted as the (quantum) Schubert cells on quantum flag manifolds. In this paper, we study the primitive ideals of Uq[w]$U_q[w]$. More precisely, it follows from the Stratification Theorem of Goodearl and Letzter, and from recent works of Mériaux–Cauchon and Yakimov, that the primitive spectrum of Uq[w]$U_q[w]$ admits a stratification indexed by those elements v∈W$v \in W$ with v≤w$v \le w$ in the Bruhat order. Moreover each stratum is homeomorphic to the spectrum of maximal ideals of a torus. The main result of this paper gives an explicit formula for the dimension of the stratum associated to a pair v≤w$v \le w$.

Abstract. For a topological abelian group X we topologize the group c0(X)$c_0(X)$ of all X -valued null sequences in a way such that when X=ℝ${X={\mathbb {R}}}$ the topology of c0(ℝ)$c_0({\mathbb {R}})$ coincides with the usual Banach space topology of the classical Banach space c 0 . If X is a non-trivial compact connected metrizable group, we prove that c0(X)$c_0(X)$ is a non-compact Polish locally quasi-convex group with countable dual group c0(X)∧$c_0(X)^{\wedge }$. Surprisingly, for a compact metrizable X , countability of c0(X)∧$c_0(X)^{\wedge }$ leads to connectedness of X . Our principal application of the above results is to the class of locally quasi-convex Mackey groups ( LQC $\rm {LQC}$-Mackey groups). A topological group (G,μ)$(G,\mu )$ from a class 𝒢$\mathcal {G}$ of topological abelian groups will be called a Mackey group in 𝒢$\mathcal {G}$ or a 𝒢$\mathcal {G}$-Mackey group if it has the following property: if ν is a group topology in G such that (G,ν)∈𝒢${(G,\nu )\in \mathcal {G}}$ and (G,ν)$(G,\nu )$ has the same character group as (G,μ)$(G,\mu )$, then ν≤μ${\nu \le \mu }$. Based upon the results obtained for c0(X)$c_0(X)$, we provide a large family of metrizable precompact (hence, locally quasi-convex) connected groups which are not LQC $\rm {LQC}$-Mackey. Namely, we show that for a connected compact metrizable group X≠{0}${X\ne \lbrace 0\rbrace }$, the group c0(X)$c_0(X)$, endowed with the topology induced from the product topology on Xℕ$X^{{\mathbb {N}}}$, is a metrizable precompact connected group which is not a Mackey group in LQC. Since metrizable locally convex spaces always carry the Mackey topology – a well-known fact from Functional Analysis –, our results prove that a Mackey theory for abelian groups is not a simple traslation of items known to hold for locally convex spaces. This paper is a contribution to the Mackey theory for groups, where properties of a topological nature like compactness or connectedness have an important role.

Abstract. A complete valued field (K,|·|)$(K, |\cdot |)$ is non-archimedean if its valuation satisfies the strong triangle inequality |a+b|≤max{|a|,|b|}$|a+b|\le \max \lbrace |a|, |b|\rbrace $ for all a,b∈K$a,b \in K$. We say that a pair ( E , F ) of Fréchet spaces over a non-archimedean field 𝕂 is splitting if for every Fréchet space G over 𝕂 and for every closed subspace D of G such that D is isomorphic to F and G / D is isomorphic to E , we can infer that the subspace D is complemented in G . In this paper we study when a pair ( E , F ) of Fréchet spaces of countable type over 𝕂 is splitting. In particular, we show that a pair (As(a),Ar(b))$(A_s(a), A_r(b))$ of power series spaces over 𝕂 is splitting if and only if s=∞$s=\infty $ or s=1$s=1$ and the set Ma,b$M_{a,b}$ of all finite limit points of the double sequence (ap/bq)p,q∈ℕ$(a_p/b_q)_{p,q\in \mathbb {N}}$ is bounded.

Abstract. A Bloch–Kato pro- p group G is a pro- p group with the property that the 𝔽p$\mathbb {F}_p$-cohomology ring of every closed subgroup of G is quadratic. It is shown that either such a pro- p group G contains no closed free pro- p groups of infinite rank, or there exists an orientation θ:G→ℤp×$\theta \colon G\rightarrow \mathbb {Z}_p^\times $ such that G is θ-abelian. In case that G is also finitely generated, this implies that G is powerful, p -adic analytic with d(G)=cd(G)$d(G)=\operatorname{cd}(G)$, and its 𝔽p$\mathbb {F}_p$-cohomology ring is an exterior algebra. These results will be obtained by studying locally powerful groups. There are certain Galois-theoretical implications, since Bloch–Kato pro- p groups arise naturally as maximal pro- p quotients and pro- p Sylow subgroups of absolute Galois groups. Finally, we study certain closure operations of the class of Bloch–Kato pro- p groups, connected with the Elementary Type Conjecture.

Abstract. In this article we prove L p estimates for resolvents of Laplace–Beltrami operators on compact Riemannian manifolds, generalizing results of Kenig, Ruiz and Sogge (1987) in the Euclidean case and Shen (2001) for the torus. We follow Sogge (1988) and construct Hadamard's parametrix, then use classical boundedness results on integral operators with oscillatory kernels related to the Carleson and Sjölin condition. Our initial motivation was to obtain L p Carleman estimates with limiting Carleman weights generalizing those of Jerison and Kenig (1985); we illustrate the pertinence of L p resolvent estimates by showing the relation with Carleman estimates. Such estimates are useful in the construction of complex geometrical optics solutions to the Schrödinger equation with unbounded potentials, an essential device for solving anisotropic inverse problems.

Abstract. The aim of this paper is to investigate the relationship between the Abelian groups G with Ext(G,G(κ))=0${\operatorname{Ext}(G,G^{(\kappa )}) = 0}$ and those satisfying Ext(G,G(κ))≅Ext(G,G)(κ)${\operatorname{Ext}(G,G^{(\kappa )})\cong \operatorname{Ext}(G,G)^{(\kappa )}}$ for various cardinals κ. In particular, we will give the structure of self-Ext-small groups. Our results answer several open problems, and give new proofs of known theorems concerning ⊕$\oplus $-splitters, tilting groups and finitely faithful S -groups.

Abstract. We generalize Ringel and Schmidmeier's theory on the Auslander–Reiten translation of the submodule category 𝒮2(A)$\mathcal {S}_2(A)$ to the monomorphism category 𝒮n(A)$\mathcal {S}_n(A)$; the category consists of all chains of (n-1)$(n-1)$ composable monomorphisms of A -modules. As in the case of n=2$n=2$, 𝒮n(A)$\mathcal {S}_n(A)$ has Auslander–Reiten sequences, and the Auslander–Reiten translation τ𝒮$\tau _{\mathcal {S}}$ of 𝒮n(A)$\mathcal {S}_n(A)$ can be explicitly formulated via τ of A -mod. Furthermore, if A is a selfinjective algebra, we study the periodicity of τ𝒮$\tau _{\mathcal {S}}$ on the objects of 𝒮n(A)$\mathcal {S}_n(A)$ and of the Serre functor F𝒮$F_{\mathcal {S}}$ on the objects of the stable monomorphism category 𝒮n(A)̲$\underline{\mathcal {S}_n(A)}$. In particular, τ𝒮2m(n+1)X≅X${\tau _{\mathcal {S}}^{2m(n+1)}X\cong X}$ for X∈𝒮n(Λ(m,t))${X\in \mathcal {S}_n(\Lambda (m, t))}$, and F𝒮m(n+1)X≅X${F_{\mathcal {S}}^{m(n+1)}X\cong X}$ for X∈𝒮n(Λ(m,t))̲${X\in \underline{\mathcal {S}_n(\Lambda (m, t))}}$, where Λ(m,t)$\Lambda (m, t)$, m≥1$m\ge 1$, t≥2$t\ge 2$, are the selfinjective Nakayama algebras.

Abstract. By elementary and direct calculations the vanishing of the (algebraic) second Lie algebra cohomology of the Witt and the Virasoro algebra with values in the adjoint module is shown. This yields infinitesimal and formal rigidity or these algebras. The first (and up to now only) proof of this important result was given 1989 by Fialowski in an unpublished note. It is based on cumbersome calculations. Compared to the original proof the presented one is quite elegant and considerably simpler.

Abstract. We consider polar unitals in projective planes over composition algebras. We prove that the little projective group of such a unital is contained in the centralizer of the polarity defining the unital. This yields information about the full automorphism group of the unital; it is contained in the automorphism group of the normal subgroup generated by all translations of the unital. Over suitable ground fields (such as the real numbers) we obtain a complete description of the automorphism group of the unital in the sense that it coincides with the centralizer of the polarity. The action of a nilpotent regular normal subgroup is used to establish isomorphisms between unitals in spaces of different dimensions, and over different fields.

Abstract. Consider the bi-harmonic differential expression of the form A=▵▵+q${A=\triangle \triangle +q}$ on a complete Riemannian manifold ( M , g ) with metric g,${g,}$ where ▵${\triangle }$ is the Laplacian on M and q≥0${q\ge 0}$ is a locally square integrable function on M . In the terminology of Everitt and Giertz, the differential expression A is said to be separated in L2(M)${L^{2}( M) }$ if for all u∈L2(M)${u\in L^{2}( M) }$ such that Au∈L2(M)${Au\in L^{2}( M) }$, we have qu∈L2(M)${ qu\in L^{2}( M)}$. In this paper we give sufficient conditions for A to be separated in L2(M)${L^{2}( M)}$.