A $B$-injector in an arbitrary finite group $G$ is defined as a maximal nilpotent subgroup of $G$, containing a subgroup $A$ of $G$ of maximal order satisfying class $(A)≤2$. The aim of this paper is to determine the $B$-injector of the covering groups of $A_n$.

The class of mixed Tsirelson spaces is an important source of examples in the recent development of the structure theory of Banach spaces. The related class of modified mixed Tsirelson spaces has also been well studied. In the present paper, we investigate the problem of comparing isomorphically the mixed Tsirelson space $T[({\mathcal{S}}_{n},\theta_{n})_{n=1}^{\infty}]$ and its modified version $T_{M}[({\mathcal{S}}_{n},\theta_{n})_{n=1}^{\infty }]$. It is shown that these spaces are not isomorphic for a large class of parameters $(θ_n)$.

The total absolute curvature of open curves in $E^3$ is discussed. We study the curves which attain the infimum of the total absolute curvature in the set of curves with fixed endpoints, end-directions, and length. We show that if the total absolute curvature of a sequence of curves in this set tends to the infimum, the limit curve must lie in a plane. Moreover, it is shown that the limit curve is either a subarc of a closed plane convex curve or a piecewise linear curve with at most three edges. The uniqueness of the curves minimizing the total absolute curvature is also discussed. This extends the results in [Yokohama Math. J. 48 (2000), 83–96], which deals with a similar problem for curves in $E^2$.

For $ν>0$, we consider the Bessel operator $S_ν$ defined on $L^{2}(ℝ^{+}, x^{2ν} dx)$ by $S_{\nu}=-\frac{d^{2}}{\,dx^{2}}-\frac{2\nu}{x}\frac{d}{dx}$. We prove, in a simple way, that the Riesz transform associated to $S_ν$ is bounded on $L^{p}(ℝ^{+}, x^{2ν}dx), 1 < p < ∞$, with a constant only depending on $p$. We also give a weighted version and estimate the constant.

To any action of a locally compact group $G$ on a pair $(A,B)$ of von Neumann algebras is canonically associated a pair $(π_{A}^{α}, π_{B}^{α})$ of unitary representations of $G$. The purpose of this paper is to provide results allowing to compare the norms of the operators $π_{A}^{α}(μ)$ and $π_{B}^{α}(μ)$ for bounded measures $μ$ on $G$. We have a twofold aim. First, to point out that several known facts in ergodic and representation theory are indeed particular cases of general results about $(π_{A}^{α}, π_{B}^{α})$. Second, under amenability assumptions, to obtain transference of inequalities that will be useful in noncommutative ergodic theory.

The concept of taut submanifold of Euclidean space is due to Carter and West, and can be traced back to the work of Chern and Lashof on immersions with minimal total absolute curvature and the subsequent reformulation of that work by Kuiper in terms of critical point theory. In this paper, we classify the reducible representations of compact simple Lie groups, all of whose orbits are tautly embedded in Euclidean space, with respect to $\mathbf{Z}_2$-coefficients.

Let $Ω$ be a bounded domain in $ℂ^n$ satisfying Catlin’s Property ($P$). Sibony has shown that $\overline{\Omega}$ possesses a Stein neighborhood basis when the boundary is of class $C^3$. In this paper, we use an alternative characterization of such domains to show that Sibony’s result holds when the boundary is of class $C^1$.

Free Markov processes are investigated in Voiculescu’s free probability theory. We show that Voiculescu’s free Markov property implies a property called “weak Markov property”, which is the classical Markov property in the commutative case; while, in the general case, the “weak Markov property” is the same as the Markov property defined by Bozejko, Kummer, and Speicher. We also show that a kind of stochastic differential equations driven by free Levy processes has solutions. The solutions are free Markov processes.

We introduce the notion of an M-family of infinite subsets of $ℕ$ which is implicitly contained in the work of Mathias. We study the structure of a pair of orthogonal hereditary families $\mathcal{A}$ and $\mathcal{B}$, where $\mathcal{A}$ is analytic and $\mathcal{B}$ is $C$-measurable and a M-family.

We obtain height estimates for compact embedded surfaces with positive constant mean curvature in a Riemannian product space $\mathbb{M}^{2}\times\mathbb{R}$ and boundary on a slice. We prove that these estimates are optimal for the homogeneous spaces $ℝ^3$, $\mathbb{S}^{2}\times\mathbb{R}$, and $ℍ^{2}×ℝ$ and we characterize the surfaces for which these bounds are achieved. We also give some geometric properties on properly embedded surfaces without boundary.

We investigate a collection of orthonormal functions that encodes information about the continued fraction expansion of real numbers. When suitably ordered these functions form a complete system of martingale differences and are a special case of a class of martingale differences considered by Gundy. By applying known results for martingales, we obtain corresponding metric theorems for the continued fraction expansion of almost all real numbers.

Let $f$ be a transcendental entire function which omits a point $a∈ℂ$. We show that if $D$ is a simply connected domain which does not contain $a$, then the full preimage $f^{−1}(D)$ is disconnected. Thus, in dynamical context, if an entire function has a completely invariant domain and omits some value, then the omitted value belongs to the completely invariant domain. We conjecture that the same property holds if $a$ is a locally omitted value (i.e., the projection of a direct singularity of $f^{−1})$. We were able to prove this conjecture for entire functions of finite order. We include some auxiliary results on singularities of $f^{−1}$ for entire functions $f$, which can be of independent interest.

In 1986, Jacobson has defined the Brauer ring $B(E, D)$ for a finite Galois field extension $E/D$, whose unit group canonically contains the Brauer group of $D$. In 1993, Cheng Xiang Chen determined the structure of the Brauer ring in the case where the extension is trivial. He revealed that if the Galois group $G$ is trivial, the Brauer ring of the trivial extension $E/E$ becomes naturally isomorphic to the group ring of the Brauer group of $E$. In this paper, we generalize this result to any finite group $G$ via the theory of the restriction functor, by means of the well-understood functor $−_+$. More generally, we determine the structure of the $F$-Burnside ring for any additive functor $F$. We construct a certain natural isomorphism of Green functors, which induces the above result with an appropriate $F$ related to the Brauer group. This isomorphism will enable us to calculate Brauer rings for some extensions. We illustrate how this isomorphism provides Green-functor-theoretic meanings for the properties of the Brauer ring shown by Jacobson, and compute the Brauer ring of the extension $ℂ/ℝ$.

We study the class of separable (real) Banach spaces $X$ which can be embedded into a space $\mathcal{C}(K)$ ($K$ compact metric) in only one way up to automorphism. We show that in addition to the known spaces $c_0$ (and all it subspaces) and $ℓ_1$ (and all its weak^*-closed subspaces) the space $c_{0}(ℓ_{1})$ has this property. We show on the other hand (answering a question of Castillo and Moreno) that $ℓ_p$ for $1 < p < ∞$ fails this property. We also show that $ℓ_p$ can be embedded in a super-reflexive space $X$ so that there is an operator $T : \ell_{p}\to\mathcal{C}(K)$ which has no extension, answering a question of Zippin.

We show that a Hilbert geometry is hyperbolic in the sense of Gromov if and only if there is an upper bound on the area of ideal triangles.