We prove that a Banach space $X$ has the Schur property if and only if every $X$-valued weakly differentiable function is Fréchet differentiable. We give a general result on the Fréchet differentiability of $f\,\circ \,T$, where $f$ is a Lipschitz function and $T$ is a compact linear operator. Finally we study, using in particular a smooth variational principle, the differentiability of the semi norm ${{\left\| {} \right\|}_{\text{lip}}}$ on various spaces of Lipschitz functions.

Let $V$ be an algebraic $\text{K}3$ surface defined over a number field $K$. Suppose $V$ has Picard number two and an infinite group of automorphisms $\mathcal{A}\,=\,\text{Aut(}V/K\text{)}$. In this paper, we introduce the notion of a vector height $\mathbf{h}:\,V\,\to \,\text{Pic(}V\text{)}\,\otimes \,\mathbb{R}$ and show the existence of a canonical vector height $\mathbf{\hat{h}}$ with the following properties:

$$\widehat{\mathbf{h}}\,\left( \sigma P \right)\,=\,{{\sigma }_{*}}\widehat{\mathbf{h}}\left( P \right)$$

$${{h}_{D}}(P)\,=\,\mathbf{\hat{h}}(P)\,\cdot \,D\,+\,O(1),$$

where $\sigma \,\in \,\mathcal{A},\,{{\sigma }_{*}}$ is the pushforward of $\sigma $ (the pullback of ${{\sigma }^{-1}}$), and ${{h}_{D}}$ is a Weil height associated to the divisor $D$. The bounded function implied by the $O(1)$ does not depend on $P$. This allows us to attack some arithmetic problems. For example, we show that the number of rational points with bounded logarithmic height in an $\mathcal{A}$-orbit satisfies

$${{N}_{\mathcal{A}(P)}}(t,\,D)\,=\,\#\{Q\,\in \,\mathcal{A}(P)\,:\,{{h}_{D}}(Q)\,<\,t\}\,=\,\frac{\mu (P)}{s\,\log \,\omega }\,\log t\,+\,O\left( \log \left( \mathbf{\hat{h}}(P)\,\cdot \,D\,+\,2 \right) \right).$$

Here, $\mu (P)$ is a nonnegative integer, $s$ is a positive integer, and $\omega $ is a real quadratic fundamental unit.

Let ${{G}_{0}}$ and ${{G}_{1}}$ be countable abelian groups. Let ${{\gamma }_{i}}$ be an automorphism of ${{G}_{i}}$ of order two. Then there exists a unital Kirchberg algebra $A$ satisfying the Universal Coefficient Theorem and with $[{{1}_{A}}]\,=\,0$ in ${{K}_{0}}(A)$, and an automorphism $\alpha \,\in \,\text{Aut}(A)$ of order two, such that ${{K}_{0}}(A)\,\cong \,{{G}_{0}}$, such that ${{K}_{1}}(A)\,\cong \,{{G}_{1}}$, and such that ${{\alpha }_{*}}\,:\,{{K}_{i}}(A)\,\to \,{{K}_{i}}(A)$ is ${{\gamma }_{i}}$. As a consequence, we prove that every ${{\mathbb{Z}}_{2}}$-graded countable module over the representation ring $R({{\mathbb{Z}}_{2}})$ of ${{\mathbb{Z}}_{2}}$ is isomorphic to the equivariant $K$-theory ${{K}^{{{\mathbb{Z}}_{2}}}}(A)$ for some action of ${{\mathbb{Z}}_{2}}$ on a unital Kirchberg algebra $A$.

Along the way, we prove that every not necessarily finitely generated $\mathbb{Z}\left[ {{\mathbb{Z}}_{2}} \right]$-module which is free as a $\mathbb{Z}$-module has a direct sum decomposition with only three kinds of summands, namely $\mathbb{Z}\left[ {{\mathbb{Z}}_{2}} \right]$ itself and $\mathbb{Z}$ on which the nontrivial element of ${{\mathbb{Z}}_{2}}$ acts either trivially or by multiplication by −1.

We describe the structure of the irreducible highest weight modules for the twisted Heisenberg–Virasoro Lie algebra at level zero. We prove that either a Verma module is irreducible or its maximal submodule is cyclic.

In this note we give examples of convex functions whose subdifferentials have unpleasant properties. Particularly, we exhibit a proper lower semicontinuous convex function on a separable Hilbert space such that the graph of its subdifferential is not closed in the product of the norm and bounded weak topologies. We also exhibit a set whose sequential normal cone is not norm closed.

In this paper, we study the modularity of certain elliptic surfaces by determining their $L$-series through their monodromy groups.

In this paper we study interpolating sequences for two related spaces of holomorphic functions in the unit ball of ${{\mathbb{C}}^{n}},\,n\,>\,1$. We first give density conditions for a sequence to be interpolating for the class ${{A}^{-\infty }}$ of holomorphic functions with polynomial growth. The sufficient condition is formally identical to the characterizing condition in dimension 1, whereas the necessary one goes along the lines of the results given by Li and Taylor for some spaces of entire functions. In the second part of the paper we show that a density condition, which for $n\,=\,1$ coincides with the characterizing condition given by Seip, is sufficient for interpolation in the (weighted) Bergman space.

This paper develops a refinement of Lasserre's algorithm for optimizing a polynomial on a basic closed semialgebraic set via semidefinite programming and addresses an open question concerning the duality gap. It is shown that, under certain natural stability assumptions, the problem of optimization on a basic closed set reduces to the compact case.

We show that if $\mathcal{A}$ is a class of ${{C}^{*}}$-algebras for which the set of formal relations $\mathcal{R}$ is weakly stable, then $\mathcal{R}$ is weakly stable for the class $B$ that contains $\mathcal{A}$ and all the inductive limits that can be constructed with the ${{C}^{*}}$-algebras in $\mathcal{A}$.

A set of formal relations $\mathcal{R}$ is said to be weakly stable for a class $\mathcal{C}$ of ${{C}^{*}}$-algebras if, in any ${{C}^{*}}$-algebra $A\,\in \,\mathcal{C}$, close to an approximate representation of the set $\mathcal{R}$ in $A$ there is an exact representation of $\mathcal{R}$ in $A$.

Let $V$ be a vector space over a field $\mathbb{K}$ of characteristic zero and ${{V}_{*}}$ be a space of linear functionals on $V$ which separate the points of $V$. We consider $V\,\otimes \,{{V}_{*}}$ as a Lie algebra of finite rank operators on $V$, and set $\mathfrak{g}\mathfrak{l}(V,\,{{V}_{*}})\,:=\,V\,\otimes \,{{V}_{*}}$. We define a Cartan subalgebra of $\mathfrak{g}\mathfrak{l}(V,{{V}_{*}})$ as the centralizer of a maximal subalgebra every element of which is semisimple, and then give the following description of all Cartan subalgebras of $\mathfrak{g}\mathfrak{l}(V,{{V}_{*}})$ under the assumption that $\mathbb{K}$ is algebraically closed. A subalgebra of $\mathfrak{g}\mathfrak{l}(V,{{V}_{*}})$ is a Cartan subalgebra if and only if it equals ${{\oplus }_{j}}({{V}_{j}}\,\otimes {{({{V}_{j}})}_{*}})\,\oplus \,({{V}^{0}}\,\otimes \,V_{*}^{0})$ for some one-dimensional subspaces ${{V}_{j}}\subseteq V$ and ${{\text{(}{{V}_{j}}\text{)}}_{*}}\subseteq {{V}_{*}}$ with ${{({{V}_{i}})}_{*}}({{V}_{j}})\,=\,{{\delta }_{ij}}\mathbb{K}$ and such that the spaces $V_{*}^{0}=\bigcap{_{j}}{{({{V}_{j}})}^{\bot }}\subseteq {{V}_{*}}$ and ${{V}^{0}}=\bigcap{_{j}}{{\left( {{({{V}_{j}})}_{*}} \right)}^{\bot }}\subseteq V$ satisfy $V_{*}^{0}({{V}^{0}})\,=\,\{0\}$. We then discuss explicit constructions of subspaces ${{V}_{j}}$ and ${{({{V}_{j}})}_{*}}$ as above. Our second main result claims that a Cartan subalgebra of $\mathfrak{g}\mathfrak{l}(V,{{V}_{*}})$ can be described alternatively as a locally nilpotent self-normalizing subalgebra whose adjoint representation is locally finite, or as a subalgebra $\mathfrak{h}$ which coincides with the maximal locally nilpotent $\mathfrak{h}$-submodule of $\mathfrak{g}\mathfrak{l}(V,{{V}_{*}})$, and such that the adjoint representation of $\mathfrak{h}$ is locally finite.

Recently [8], a harmonic theory was developed for a compact contact manifold from the viewpoint of the transversal geometry of contact flow. A contact flow is a typical example of geodesible flow. As a natural generalization of the contact flow, the present paper develops a harmonic theory for various flows on compact manifolds. We introduce the notions of $H$-harmonic and ${{H}^{*}}$-harmonic spaces associated to a Hörmander flow. We also introduce the notions of basic harmonic spaces associated to a weak basic flow. One of our main results is to show that in the special case of isometric flow these harmonic spaces are isomorphic to the cohomology spaces of certain complexes. Moreover, we find an obstruction for a geodesible flow to be isometric.

We prove a quantized version of a theorem by M. V. Sheĭnberg: A uniform algebra equipped with its canonical, i.e., minimal, operator space structure is operator amenable if and only if it is a commutative ${{C}^{*}}$-algebra.