We exhibit examples of $F$-spaces with trivial dual which are isomorphic to its quotient by a line, thus solving a problem in Rolewicz's Metric Linear Spaces.

An $\text{AF}$ flow is a one-parameter automorphism group of an $\text{AF}$${{C}^{*}}$-algebra $A$ such that there exists an increasing sequence of invariant finite dimensional sub-${{C}^{*}}$-algebras whose union is dense in $A$. In this paper, a classification of ${{C}^{*}}$-dynamical systems of this form up to equivariant isomorphism is presented. Two pictures of the actions are given, one in terms of a modified Bratteli diagram/pathspace construction, and one in terms of a modified ${{K}_{0}}$ functor.

We carry out the computation of the Iwasawa invariants $\rho _{S}^{T},\,\mu _{S}^{T},\,\lambda _{S}^{T}$ associated to abelian $T$-ramified $S$-decomposed $\ell$-extensions over the finite steps ${{K}_{n}}$ of the cyclotomic ${{\mathbb{Z}}_{\ell }}$ -extension ${{K}_{\infty }}/K$ of a number field of CM-type.

Let ${{\left\{ {{A}_{t}} \right\}}_{t>0}}$ be the dilation group in ${{\mathbb{R}}^{n}}$ generated by the infinitesimal generator $M$ where ${{A}_{t}}\,=\,\exp \left( M\,\log \,t \right)$, and let $\varrho \,\in \,{{C}^{\infty }}\left( \mathbb{R}{{}^{n\,}}\backslash \,\left\{ 0 \right\} \right)$ be a ${{A}_{t}}$-homogeneous distance function defined on ${{\mathbb{R}}^{n}}$. For $f\,\in \,\mathfrak{S}\left( \mathbb{R}{{}^{n}} \right)$, we define the maximal quasiradial Bochner-Riesz operator $\mathfrak{M}_{\varrho }^{\delta }$ of index $\delta \,>\,0$ by

$$\mathfrak{M}_{\varrho }^{\delta }f\left( x \right)\,=\,\underset{t>0}{\mathop{\sup }}\,\left| {{\mathcal{F}}^{-1}}\left[ \left( 1-{\varrho }/{t}\; \right)_{+}^{\delta }\hat{f} \right]\left( x \right) \right|.$$

If ${{A}_{t\,}}=\,tI$ and $\left\{ \xi \,\in \,{{\mathbb{R}}^{n\,}}\,|\,\varrho \left( \xi \right)\,=\,1 \right\}$ is a smooth convex hypersurface of finite type, then we prove in an extremely easy way that $\mathfrak{M}_{\varrho }^{\delta }$ is well defined on ${{H}^{p}}\left( {{\mathbb{R}}^{n}} \right)$ when $\delta \,=\,n(1/p\,-\,1/2)\,-\,1/2$ and $0\,<\,p\,<\,1$; moreover, it is a bounded operator from ${{H}^{p}}\left( {{\mathbb{R}}^{n}} \right)$ into ${{L}^{p,\infty }}\left( {{\mathbb{R}}^{n}} \right)$ .

If ${{A}_{t}}\,=\,tI$ and $\varrho \,\in \,{{C}^{\infty }}\left( \mathbb{R}{{}^{n\,}}\backslash \,\left\{ 0 \right\} \right)$ , we also prove that $\mathfrak{M}_{\varrho }^{\delta }$ is a bounded operator from ${{H}^{p}}\left( {{\mathbb{R}}^{n}} \right)$ into ${{L}^{p}}\,\left( {{\mathbb{R}}^{n}} \right)$ when $\delta \,>\,n(1/p\,-\,1/2)\,-\,1/2$ and $0\,<\,p\,<\,1$.

Suppose we are given a finite-dimensional vector space $V$ equipped with an $F$-rational action of a linearly algebraic group $G$, with $F$ a characteristic zero field. We conjecture the following: to each vector $v\,\in \,V(F)$ there corresponds a canonical $G(F)$-orbit of semisimple vectors of $V$. In the case of the adjoint action, this orbit is the $G(F)$-orbit of the semisimple part of $v$, so this conjecture can be considered a generalization of the Jordan decomposition. We prove some cases of the conjecture.

Let $H$ be a complex Hilbert space, and $\mathcal{H}\left( H \right)$ be the real linear space of bounded selfadjoint operators on $H$. We study linear maps $\phi :\,\mathcal{H}(H)\,\to \,\mathcal{H}(H)$ leaving invariant various properties such as invertibility, positive definiteness, numerical range, etc. The maps $\phi$ are not assumed a priori continuous. It is shown that under an appropriate surjective or injective assumption $\phi$ has the form $X\mapsto \xi TX{{T}^{*}}orX\mapsto \xi T{{X}^{t}}{{T}^{*}}$, for a suitable invertible or unitary $T$ and $\xi \,\in \,\{1,\,-1\}$, where ${{X}^{t}}$ stands for the transpose of $X$ relative to some orthonormal basis. Examples are given to show that the surjective or injective assumption cannot be relaxed. The results are extended to complex linear maps on the algebra of bounded linear operators on $H$. Similar results are proved for the (real) linear space of (selfadjoint) operators of the form $\alpha I\,+\,K$, where $\alpha$ is a scalar and $K$ is compact.

Recently there has been tremendous interest in counting the number of integral points in $n$-dimensional tetrahedra with non-integral vertices due to its applications in primality testing and factoring in number theory and in singularities theory. The purpose of this note is to formulate a conjecture on sharp upper estimate of the number of integral points in $n$-dimensional tetrahedra with non-integral vertices. We show that this conjecture is true for low dimensional cases as well as in the case of homogeneous $n$-dimensional tetrahedra. We also show that the Bernoulli polynomials play a role in this counting.

We study the dimension of “random” Euclidean sections of direct sums of normed spaces. We compare the obtained results with results from $[\text{LMS}]$, to show that for the direct sums the standard randomness with respect to the Haar measure on Grassmanian coincides with a much “weaker” randomness of “diagonal” subspaces (Corollary 1.4 and explanation after). We also add some relative information on “phase transition”.

This paper shows that some characterizations of minimally thin sets connected with a domain having smooth boundary and a half-space in particular also hold for the minimally thin sets at a corner point of a special domain with corners, i.e., the minimally thin set at $\infty$ of a cone.

Let $M$ be a compact, connected, orientable, irreducible 3-manifold with a torus boundary. It is known that if two Dehn fillings on $M$ along the boundary produce a reducible manifold and a manifold containing a Klein bottle, then the distance between the filling slopes is at most three. This paper gives a remarkably short proof of this result.

Let $G$ be a finitely generated, infinite group, let $p\,>\,1$, and let ${{L}^{p}}\left( G \right)$ denote the Banach space $\left\{ \sum{_{x\in G}{{a}_{x}}x}|\sum{_{x\in G}|{{a}_{x}}{{|}^{p}}<\infty } \right\}$. In this paper we will study the first cohomology group of $G$ with coefficients in ${{L}^{p}}\left( G \right)$, and the first reduced ${{L}^{p}}$-cohomology space of $G$. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups.

This paper studies the following question: Given an ${\omega }'$-symplectic action of a Lie group on a manifold $M$ which coincides, as a smooth action, with a Hamiltonian $\omega$-action, when is this action a Hamiltonian ${\omega }'$-action? Using a result of Morse-Bott theory presented in Section 2, we show in Section 3 of this paper that such an action is in fact a Hamiltonian ${\omega }'$-action, provided that $M$ is compact and that the Lie group is compact and connected. This result was first proved by Lalonde-McDuff-Polterovich in 1999 as a consequence of a more general theory that made use of hard geometric analysis. In this paper, we prove it using classical methods only.

The purpose of this note is to extend to an arbitrary generalized Hopf and Calabi-Eckmann manifold the following result of Kalyan Mukherjea: Let ${{V}_{n}}={{\mathbb{S}}^{2n+1}}\times {{\mathbb{S}}^{2n+1}}$ denote a Calabi-Eckmann manifold. If $f,g:\,{{V}_{n}}\to {{\mathbb{P}}^{n}}$ are any two holomorphic maps, at least one of them being non-constant, then there exists a coincidence: $f(x)\,=\,g(x)$ for some $x\in {{V}_{n}}$. Our proof involves a coincidence theorem for holomorphic maps to complex projective varieties of the form $G/P$ where $G$ is complex simple algebraic group and $P\,\subset \,G$ is a maximal parabolic subgroup, where one of the maps is dominant.

We show here that the commutator subgroup of a free group of finite rank poses a basis of Bachmuth's type.

The behaviour of the Hasse-Schmidt algebra of higher derivations under localization is studied using André cohomology. Elementary techniques are used to describe the Hasse-Schmidt derivations on certain monomial rings in the nonmodular case. The localization conjecture is then verified for all monomial rings.

Upper bounds of second order Dehn functions of asynchronously automatic groups are obtained.