Nous démontrons que tous les plongements d’une variété compacte sans bord et simplement connexe de dimension quatre dans la sphère de dimension six sont concordants.

We construct several examples of higher-dimensional Calabi–Yau manifolds and prove their modularity.

Let $v\,\ge \,k\,\ge \,1$ and $\lambda \,\ge \,0$ be integers. A block design$\text{BD}\left( v,\,k,\,\lambda \right)$ is a collection $\mathcal{A}$ of $k$-subsets of a $v$-set $X$ in which every unordered pair of elements from $X$ is contained in exactly $\lambda $ elements of $\mathcal{A}$. More generally, for a fixed simple graph $G$, a graph design$\text{GD}\left( v,\,G,\,\lambda \right)$ is a collection $\mathcal{A}$ of graphs isomorphic to $G$ with vertices in $X$ such that every unordered pair of elements from $X$ is an edge of exactly $\lambda $ elements of $\mathcal{A}$. A famous result of Wilson says that for a fixed $ $ and $\lambda $, there exists a $\text{GD}\left( v,\,G,\,\lambda \right)$ for all sufficiently large $ $ satisfying certain necessary conditions. A block (graph) design as above is resolvable if $\mathcal{A}$ can be partitioned into partitions of (graphs whose vertex sets partition) $X$. Lu has shown asymptotic existence in $v$ of resolvable $\text{BD}\left( v,\,k,\,\lambda \right)$, yet for over twenty years the analogous problem for resolvable $\text{GD}\left( v,\,G,\,\lambda \right)$ has remained open. In this paper, we settle asymptotic existence of resolvable graph designs.

It is shown that Schatten $p$-classes of operators between Hilbert spaces of different (infinite) dimensions have ultrapowers which are (completely) isometric to non-commutative ${{L}_{p}}$-spaces. On the other hand, these Schatten classes are not themselves isomorphic to non-commutative ${{L}_{p}}$ spaces. As a consequence, the class of non-commutative ${{L}_{p}}$-spaces is not axiomatizable in the first-order language developed by Henson and Iovino for normed space structures, neither in the signature of Banach spaces, nor in that of operator spaces. Other examples of the same phenomenon are presented that belong to the class of corners of non-commutative ${{L}_{p}}$-spaces. For $p\,=\,1$ this last class, which is the same as the class of preduals of ternary rings of operators, is itself axiomatizable in the signature of operator spaces.

If $A$ is a subset of the set of reflections of a finite Coxeter group $W$, we define a sub-$\mathbb{Z}$-module ${{\mathcal{D}}_{A}}\left( W \right)$ of the group algebra $\mathbb{Z}W$. We discuss cases where this submodule is a subalgebra. This family of subalgebras includes strictly the Solomon descent algebra, the group algebra and, if $W$ is of type $B$, the Mantaci–Reutenauer algebra.

The paper offers a study of the inverse Laplace transforms of the functions ${{I}_{n}}\left( rs \right)\,{{\{sI_{n}^{\prime }\,\left( s \right)\}}^{-1}}$ where ${{I}_{n}}$ is the modified Bessel function of the first kind and $r$ is a parameter. The present study is a continuation of the author's previous work on the singular behavior of the special case of the functions in question, $r=1$. The general case of $r\,\in \,\left[ 0,\,1 \right]$ is addressed, and it is shown that the inverse Laplace transforms for such $r$ exhibit significantly more complex behavior than their predecessors, even though they still only have two different types of points of discontinuity: singularities and finite discontinuities. The functions studied originate from non-stationary fluid-structure interaction, and as such are of interest to researchers working in the area.

In this paper we show that any Frobenius split, smooth, projective threefold over a perfect field of characteristic $p\,>\,0$ is Hodge–Witt. This is proved by generalizing to the case of threefolds a well-known criterion due to $\text{N}$. Nygaard for surfaces to be Hodge-Witt. We also show that the second crystalline cohomology of any smooth, projective Frobenius split variety does not have any exotic torsion. In the last two sections we include some applications.

For $p\,>\,0$ and for a given set $E$ of type ${{G}_{\delta }}$ in the boundary of the unit disc $\partial \mathbb{D}$ we construct a holomorphic function $f\,\in \,\mathbb{O}\left( \mathbb{D} \right)$ such that

$${{\int_{\mathbb{D}\backslash \left[ 0,\,1 \right]E}{\left| f \right|}}^{p}}\,d{{\mathfrak{L}}^{2}}\,<\,\infty \,\text{and}\,E\,=\,{{E}^{p}}\left( f \right)\,=\,\{\,z\,\in \,\partial \mathbb{D}\,:\,\int _{0}^{1}\,{{\left| f\left( tz \right) \right|}^{p}}\,dt\,=\,\infty \}.$$

In particular if a set $E$ has a measure equal to zero, then a function $f$ is constructed as integrable with power $p$ on the unit disc $\mathbb{D}$.

Let $p$ be a prime and $F$ a field containing a primitive $p$-th root of unity. Then for $n\,\in \,\mathbb{N}$, the cohomological dimension of the maximal pro-$p$-quotient $G$ of the absolute Galois group of $F$ is at most $n$ if and only if the corestriction maps ${{H}^{n}}\left( H,\ {{\mathbb{F}}_{p}} \right)\,\to \,{{H}^{n}}\left( G,\ {{\mathbb{F}}_{p}} \right)$ are surjective for all open subgroups $H$ of index $p$. Using this result, we generalize Schreier's formula for ${{\dim}_{{{\mathbb{F}}_{p}}}}\,{{H}^{1}}\,\left( H,\ {{\mathbb{F}}_{p}} \right)$ to ${{\dim}_{{{\mathbb{F}}_{p}}}}{{H}^{n}}\left( H,\ {{\mathbb{F}}_{p}} \right)$.

Soit $q$ une puissance d’un nombre premier $p$. Dans cette note on établit la généralisation suivante d’un théorème de Wintenberger : tout sous-groupe abélien fermé du groupe des ${{\mathbb{F}}_{q}}$-automorphismes continus du corps des séries formelles ${{\mathbb{F}}_{q}}((X\text{))}$ muni de sa filtration de ramification est un groupe filtré isomorphe au groupe de Galois d’une extension abélienne d’un corps local à corps résiduel ${{\mathbb{F}}_{q}}$, filtré par les groupes de ramification de l’extension en numérotation inférieure.

Let $R$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $R$ and $M$ a finitely generated $R$-module. Let $t$ be a non-negative integer. It is known that if the local cohomology module $\text{H}_{\mathfrak{a}}^{i}\,\left( M \right)$ is finitely generated for all $i\,<\,t$, then $\text{Ho}{{\text{m}}_{R}}\,\left( R/\mathfrak{a},\,\text{H}_{\mathfrak{a}}^{t}\,\left( M \right) \right)$ is finitely generated. In this paper it is shown that if $\text{H}_{\mathfrak{a}}^{i}\,\left( M \right)$ is Artinian for all $i\,<\,t$, then $\text{Ho}{{\text{m}}_{R}}\,\left( R/\mathfrak{a},\,\text{H}_{\mathfrak{a}}^{t}\,\left( M \right) \right)$ need not be Artinian, but it has a finitely generated submodule $N$ such that $\text{Ho}{{\text{m}}_{R}}\left( R/\mathfrak{a},\text{H}_{\mathfrak{a}}^{t}\left( M \right) \right)/N$ is Artinian.

Let $\mathfrak{g}$ be a semisimple complex Lie algebra and $\mathfrak{k}\,\subset \mathfrak{g}$ be any algebraic subalgebra reductive in $\mathfrak{g}$. For any simple finite dimensional $\mathfrak{k}$-module $V$, we construct simple $\left( \mathfrak{g},\mathfrak{k} \right)$-modules $M$ with finite dimensional $\mathfrak{k}$-isotypic components such that $V$ is a $\mathfrak{k}$-submodule of $M$ and the Vogan norm of any simple $\mathfrak{k}$-submodule $V\prime \subset M,V\prime \ne \,V$, is greater than the Vogan norm of $V$. The $\left( \mathfrak{g},\mathfrak{k} \right)$-modules $M$ are subquotients of the fundamental series of $\left( \mathfrak{g},\mathfrak{k} \right)$-modules.

We present partial positive results supporting a conjecture that admitting an equivalent Lipschitz (or uniformly) weak* Kadec–Klee norm is a three space property.

It is shown that if a Banach space is saturated with infinite dimensional subspaces in which all “special” $n$-tuples of vectors are equivalent with constants independent of $n$-tuples and of $n$, then the space contains asymptotic-${{l}_{p}}$ subspaces for some $1\,\le \,p\,\le \,\infty $. This extends a result by Figiel, Frankiewicz, Komorowski and Ryll-Nardzewski.

Let $G$ be a compact topological group and let $f:\,G\,\to \,G$ be a continuous transformation of $G$. Define ${{f}^{*}}:G\to G$ by ${{f}^{*}}\left( x \right)\,=\,f\left( {{x}^{-1}} \right)x$ and let $\mu \,=\,{{\mu }_{G}}$ be Haar measure on $G$. Assume that $H\,=\,\text{IM}\,{{f}^{*}}$ is a subgroup of $G$ and for every measurable $C\,\subseteq \,H,\,{{\mu }_{G}}{{\left( \left( {{f}^{*}} \right) \right)}^{-1}}\left( \left( C \right) \right)\,=\,\mu H\left( C \right)$. Then for every measurable $C\,\subseteq \,G$, there exist $S\subseteq C$ and $g\,\in \,G$ such that $f\left( S{{g}^{-1}} \right)\,\subseteq \,C{{g}^{-1}}$ and $\mu \left( S \right)\,\ge \,{{\left( \mu \left( C \right) \right)}^{2}}$.