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 $X,\,Y$ be reduced and irreducible compact complex spaces and $S$ the set of all isomorphism classes of reduced and irreducible compact complex spaces $W$ such that $X\,\times \,Y\,\cong \,X\,\times \,W$. Here we prove that $S$ is at most countable. We apply this result to show that for every reduced and irreducible compact complex space $X$ the set $S(X)$ of all complex reduced compact complex spaces $W$ with $X\,\times \,{{X}^{\sigma }}\,\cong \,W\,\times \,{{W}^{\sigma }}$ (where ${{A}^{\sigma }}$ denotes the complex conjugate of any variety $A$) is at most countable.

An integral domain $D$ with identity is condensed (resp., strongly condensed) if for each pair of ideals $I,\,J$ of $D,\,IJ\,=\,\{ij\,;\,i\,\in I,j\in J\,\}$ (resp., $IJ=iJ$ for some $i\,\in \,I$ or $IJ\,=Ij$ for some $j\,\in \,J$). We show that for a Noetherian domain $D,\,D$ is condensed if and only if $\text{Pic}\left( D \right)\,=0$ and $D$ is locally condensed, while a local domain is strongly condensed if and only if it has the two-generator property. An integrally closed domain $D$ is strongly condensed if and only if $D$ is a Bézout generalized Dedekind domain with at most one maximal ideal of height greater than one. We give a number of equivalencies for a local domain with finite integral closure to be strongly condensed. Finally, we show that for a field extension $k\,\subseteq K$, the domain $D=\,k+XK[[X]]$ is condensed if and only if $[K:k]\,\le \,2$ or $[K:k]\,=\,3$ and each degree-two polynomial in $k[X]$ splits over $k$, while $D$ is strongly condensed if and only if $[K:k]\,\le \,2$.

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.

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.

Recent papers have shown that ${{C}^{1}}$ maps $F:\,{{\mathbb{R}}^{2}}\,\to {{\mathbb{R}}^{2}}$ whose Jacobians have constant eigenvalues can be completely characterized if either the eigenvalues are equal or $F$ is a polynomial. Specifically, $F\,=\,(u,\,v)$ must take the form

$$u\,=\,ax\,+\,by\,+\,\beta \phi (\alpha x\,+\,\beta y)\,+\,e$$

$$v\,=\,cx\,+\,dy\,-\,\alpha \phi \,(\alpha x\,+\,\beta y)\,+\,f$$

for some constants $a,\,b,\,c,\,d,\,e,\,f,\,\alpha ,\,\beta $ and a ${{C}^{1}}$ function $\phi $ in one variable. If, in addition, the function $\phi $ is not affine, then

1

$$\alpha \beta (d\,-\,a)\,+\,b{{\alpha }^{2}}\,-\,c{{\beta }^{2}}\,=\,0.$$

This paper shows how these theorems cannot be extended by constructing a real-analytic map whose Jacobian eigenvalues are $\pm 1/2$ and does not fit the previous form. This example is also used to construct non-obvious solutions to nonlinear PDEs, including the Monge—Ampère equation.

We study group algebras $FG$ which can be graded by a finite abelian group $\Gamma $ such that $FG$ is $\beta $-commutative for a skew-symmetric bicharacter $\beta $ on $\Gamma $ with values in ${{F}^{*}}$.

Soit $K$ un corps de nombres de degré sur $\mathbb{Q}$ inférieur ou égal à 2. On se propose dans ce travail de faire quelques remarques sur la question de l'existence de deux éléments non nuls $a$ et $b$ de $K$, et d'un entier $n\,\ge \,4$, tels que l'équation $a{{x}^{n}}\,+\,b{{y}^{n\,}}=\,1$ possède au moins trois points distincts non triviaux. Cette étude se ramène à la recherche de points rationnels sur $K$ d'une variété projective dans ${{\mathbb{P}}^{5}}$ de dimension 3, ou d'une surface de ${{\mathbb{P}}^{3}}$.

We generalize the well-known result that a square traceless complex matrix is unitarily similar to a matrix with zero diagonal to arbitrary connected semisimple complex Lie groups $G$ and their Lie algebras $\mathfrak{g}$ under the action of a maximal compact subgroup $K$ of $G$. We also introduce a natural partial order on $\mathfrak{g}:\,x\,\le y$ if $f(K\,\cdot \,x)\,\subseteq \,f(K\,\cdot \,y)$ for all $f\,\in \,{{\mathfrak{g}}^{*}}$, the complex dual of $\mathfrak{g}$. This partial order is $K$-invariant and induces a partial order on the orbit space $\mathfrak{g}/K$. We prove that, under some restrictions on $\mathfrak{g}$, the set $f(K\,\cdot \,x)$ is star-shaped with respect to the origin.

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 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.

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.

Let $d\,>\,1$ be a square-free integer. Power residue criteria for the fundamental unit ${{\varepsilon }_{d}}$ of the real quadratic fields $\mathbb{Q}(\sqrt{d})$ modulo a prime $p$ (for certain $d$ and $p$) are proved by means of class field theory. These results will then be interpreted as criteria for the solvability of the negative Pell equation ${{x}^{2}}\,-\,d{{p}^{2}}{{y}^{2}}\,=\,-1$. The most important solvability criterion deals with all $d$ for which $\mathbb{Q}(\sqrt{-d})$ has an elementary abelian 2-class group and $p\,\equiv \,5$ (mod 8) or $p\,\equiv \,9$ (mod 16).

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 $0\,<\,p\,<\,1$.

Let $q\,=\,{{p}^{r}}$ with $p$ an odd prime, and ${{\mathbf{F}}_{q}}$ denote the finite field of $q$ elements. Let $\text{Tr}\,:\,{{\mathbf{F}}_{q}}\,\to \,{{\mathbf{F}}_{p}}$ be the usual trace map and set ${{\zeta }_{p}}\,=\,\exp (2\pi i/p)$. For any positive integer $e$, define the (modified) Gauss sum ${{g}_{r}}(e)$ by

$${{g}_{r}}(e)=\underset{x\in {{\mathbf{F}}_{q}}}{\mathop{\sum }}\,\zeta _{p}^{\text{Tr}{{x}^{e}}}$$

Recently, Evans gave an elegant determination of ${{g}_{1}}(12)$ in terms of ${{g}_{1}}(3),\,{{g}_{1}}(4)$ and ${{g}_{1}}(6)$ which resolved a sign ambiguity present in a previous evaluation. Here I generalize Evans' result to give a complete determination of the sum ${{g}_{r}}(12)$.

Let $U(n)$ be the group of $n\,\times \,n$ unitary matrices. We show that if $\phi $ is a linear transformation sending $U(n)$ into $U(m)$, then $m$ is a multiple of $n$, and $\phi $ has the form

$$A\,\mapsto \,V[(A\,\otimes \,{{I}_{s}})\,\otimes \,({{A}^{t}}\,\otimes \,{{I}_{r}})]W$$

for some $V,\,W\,\in \,U(m)$. From this result, one easily deduces the characterization of linear operators that map $U(n)$ into itself obtained by Marcus. Further generalization of the main theorem is also discussed.

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.

We give an algorithm for a surgery description of a $p$-fold cyclic branched cover of ${{B}^{3}}$ branched along a tangle. We generalize constructions of Montesinos and Akbulut-Kirby.

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.