We study the topological properties of cohomogeneity one flat manifolds and their orbits. Among other results we prove that principal orbits of R^n are isometric to R^{n-1} or S^k(c)\times R^{n-k-1}. We show that if M has one singular orbit, it is a totally geodesic submanifold of M and if M is orientable then there is at most one singular orbit.

The purpose of this paper is to study an optimal control problem associated with the thermally coupled Navier-Stokes equations. An existence result for this problem is obtained. The most important result of this paper is the proof of the existence and regularity of a solution of the adjoint system. By defining several functions, this system (which is not a divergence free one) is replaced by a divergence free system.

In this paper, relative pictures are used to analyze a certain family of exponent-sum two equations over groups.

Existence theorems for the equation F(x)=\varphi (x) are proved when F is a function with “good” surjective properties and \varphi satisfies certain compactness conditions on countable subsets of the space. Also results for certain homotopic perturbations of the equation are obtained. The results lead to various fixed point theorems of Darbo type for F=id, but they are also applicable if F acts between different spaces. Also the inclusions F(x)\in \varphi (x) (resp. F(x)\subseteq \varphi (x)) for multivalued functions \varphi (resp. F and \varphi) are studied. There are some connections with the theory of 0-epi maps.

We discuss classes of topological groups which can be approximated by p-adic Lie groups, and varieties of Hausdorff groups generated by classes of \hboxp-adic Lie groups (for a single or multiple p). We give several characterizations of locally compact pro-p-adic Lie groups and locally compact pro-discrete groups, and prove a “pro-version” of Cartan's Theorem: whenever a locally compact group is a pro-p-adic Lie group and a pro-q-adic Lie group for distinct primes p and q, it is pro-discrete. If a locally compact group can be approximated by p-adic Lie groups for variable primes p, then it is a pro-p-adic Lie group for some prime p.

We apply a previously obtained ansatz for extremal Kähler metrics to show that if a manifold admits a Hodge metric with constant scalar curvature, then the total space of the projectivization of a line bundle with first Chern class equal to the Kähler class of the metric admits a one-parameter family of extremal Kähler metrics. This generalizes earlier constructions.

Let Ea(u,v) be the extremal algebra determined by two hermitians u and v with u^2=v^2=1. We show that: Ea(u,v)={f+gu:f,g\in C(𝕋)} , where [ ] is the unit circle; Ea(u,v) is C^*-equivalent to C^*({\cal G}), where {\cal G} is the infinite dihedral group; most of the hermitian elements k of Ea(u,v) have the property that k^n is hermitian for all odd n but for no even n; any two hermitian words in {\cal G} generate an isometric copy of Ea(u,v) in Ea(u,v).

The Kadets path distance between Banach spaces X and Y is defined to be the infimum of the lengths with respect to the Kadets distance of all curves joining X and Y. If there is no curve joining X and Y, the Kadets path distance between X and Y is defined to be infty .

Some approaches to estimates of the Kadets path distance from above and from below are developed. In particular, the Kadets path distances between the spaces l_p^n,\ p\in [1,+\rm (inf)ty ], n\in {\b (N)} are estimated.

Let B(H) denote the C*-algebra of all bounded linear operators on a separable Hilbert space H. For A,B∈B(H), the chordal transform f_(A,B), as an operator on B(H), is defined by {f_(A,B)(X)=(\vert A^*\vert ^2+I)^(-1/2){\delta_(A,B)(X)(\vert B\vert ^2+I)^(-1/2)}, where {\delta_(A,B)} is the generalized derivation defined on B(H) by {\delta _(A,B)(X)=AX-XB}. Orthogonality of the range and the kernel of f_(A,B), with respect to the unitarily invariant norms \vert \vert \vert .\vert \vert \vert , are discussed. It is shown that if A, B are self-adjoint, then {\vert \vert \vert f_(A,B)(X)\vert \vert \vert \le \vert \vert \vert X\vert \vert \vert for all X. Related norm inequalities comparing f_(A,B) and {\delta _(A,B) are also given.

We calculate the height of quantum determinantal ideals in the algebra of quantum matrices, and also the Gelfand-Kirillov, Krull and Classical Krull dimensions of the corresponding quantum determinantal factors.

In this paper we prove that the ideal property and the projection property do not conincide in general even in the separable case (despite the fact that, as we proved before, they are the same for GAH algebras-and, in particular, for AH algebras-and for separable LB algebras). We also study the behaviour of the projection property with respect to several natural operations.

We study the conditions which force a semiperfect ring to admit a Nakayama permutation of its basic idempotents. We also give a few necessary and sufficient conditions for a semiperfect ring R, which cogenerates every 2-generated right R-module, to be right pseudo-Frobenius.

A Banach space is called (almost) transitive if the isometry group acts (almost) transitively on the unit sphere. The main problems around transitivity are the Banach-Mazur conjecture that the only separable and transitive Banach spaces are the Hilbert ones (1930) and the Wood conjecture that C_0(L) cannot be almost transitive in its natural supremum norm unless L is a singleton (1982). In this note we give necessary and sufficient conditions on the locally compact space L for the (almost) transitivity of C_0(L). This will clarify the topological content of Wood's problem.

We prove that the module categories of Noether algebras (i.e., algebras module finite over a noetherian center) and affine noetherian PI algebras over a field enjoy the following product property: whenever a direct product \prod _(n \in ℕ) M_n of finitely generated indecomposable modules M_n is a direct sum of finitely generated objects, there are repeats among the isomorphism types of the M_n. The rings with this property satisfy the pure semisimplicity conjecture which stipulates that vanishing one-sided pure global dimension entails finite representation type.

In this paper, we show that Weyl's theorem holds for operators having the single valued extension property and quasisimilarity preserves Weyl's theorem for these operators under some assumptions for spectral subsets, respectively.

AC-operators generalise normal operators on Hilbert space in the context of well-boundedness. In this paper we study AC-operators T=U+iV, where U and V are commuting well-bounded operators with decomposition of the identity of bounded variation. We also explore some properties of AC-operators by applying the theory of (Foiaş) decomposable operators.

This paper considers the long-time behavior of solutions for the Cauchy problem of a class of second-order nonlinear differential equations: -x''+ f(t,x,x')x'+g(x)=h(t). Under appropriate conditions it is shown that the solutions of the problem possess some dichotomy properties.

If R is a complete discrete valuation ring and M is a reduced, torsion-free R-module of rank \kappa, where \aleph_0 \leq \kappa < 2^ (\aleph_0), we show that M \prop\oplus_(\aleph_0) R \oplus C for some R-module C. As a consequence, it must be the case that M \prop M \oplus (\oplus{_\alpha}R), where \alpha \leq \aleph_0, and {\rm (End)_R}M/\rm (Fin)M has rank at least 2^ (\aleph_0), where Fin M denotes the set of endomorphisms of M with finite rank image.