We present a classification of the so-called “additive symmetric 2-cocycles” of arbitrary degree and dimension over $\mathbb{F}_p$, along with a partial result and some conjectures for $m$-cocycles over $\mathbb{F}_p$, $m \gt 2$. This expands greatly on a result originally due to Lazard and more recently investigated by Ando, Hopkins and Strickland, and together with their work this culminates in a complete classification of $2$-cocycles over an arbitrary commutative ring. The ring classifying these polynomials finds application in algebraic topology, to be fully explored in a sequel.

We study qualitative properties of the operator-\break weighted composition maps ${W_{\psi,\varphi}} : f\mapsto\psi(f\circ\varphi)$ on the vector-valued spaces $H^\infty_v(X)$ of $X$-valued analytic functions $f : {\mathbb{D}}\to X$, where ${\mathbb{D}}$ is the unit disk, $X$ is a complex Banach space, $\varphi$ is an analytic self-map of ${\mathbb{D}}$, $\psi$ is an analytic operator-valued function on ${\mathbb{D}}$, and $v$ is a bounded continuous weight on ${\mathbb{D}}$. Boundedness and compactness properties of ${W_{\psi,\varphi}}$ are characterized on $H^\infty_v(X)$ for infinite-dimensional $X$. It turns out that the (weak) compactness of ${W_{\psi,\varphi}}$ also involves properties of the auxiliary operator $T_\psi : x \mapsto\psi(\cdot)x$ from $X$ to $H^\infty_v(X)$, in contrast to the familiar scalar-valued setting $X = \mathbb C$.

We give criteria for a function to be in the Hardy space on a bounded complete Reinhardt domain. Using these and known one-dimensional results, we obtain boundedness conditions for Hausdorff operators on Hardy spaces in Reinhardt domains. The only known earlier result for the polydisk is a paticular case of the obtained results.

We study the projective embeddings of complete Gorenstein toric del Pezzo surfaces by ample complete linear systems, especially of minimal degree and dimension. Complete Gorenstein toric del Pezzo surfaces are in one-to-one correspondence with the 2-dimensional reflexive integral convex polytopes, which are classified into 16 types up to isomorphisms of lattices. Our main result shows that the minimal dimension and the minimal degree of all the ample complete linear systems on such a surface are attained by the primitive anti-canonical class except one case. From this, we determine the projective embeddings of these surfaces which are global complete intersections. We also show that the minimal free resolution of the defining ideal of the image under the anti-canonical embedding of these surfaces is given by an Eagon–Northcott complex.

In earlier work, the authors have extended Nehari’s well-known Schwarzian derivative criterion for univalence of analytic functions to a univalence criterion for canonical lifts of harmonic mappings to minimal surfaces. The present paper develops some quantitative versions of that result in the form of two-point distortion theorems. Along the way some distortion theorems for curves in ℝⁿ are given, thereby recasting a recent injectivity criterion of Chuaqui and Gevirtz in quantitative form.

Nonsymmetrically weighted extensions of the Balian--Low theorem are proved for Gabor systems $\mathcal{G}(g,1,1)$ that are complete and minimal in ${L^2(\mathbb{R})}$. For $g\in{L^2(\mathbb{R})}$, it is proved that if $3 \lt p \leq4 \leq q \lt \infty$ satisfy $3/p + 1/q = 1$ and $\int|x|^p |g(x)|^2 \, dx \lt \infty$ and $\int|\xi|^q |\widehat{g}(\xi)|^2 \, d\xi \lt \infty$ then $\mathcal{G}(g,1,1) = \{e^{2\pi i n x} g(x-k)\}_{k,n \in{\mathbb{Z}}}$ cannot be complete and minimal in ${L^2(\mathbb{R})}$. For the endpoint case $(p,q)=(3,\infty)$, it is proved that if $g\in{L^2(\mathbb{R})}$ is compactly supported and $\int|\xi|^3 |\widehat{g}(\xi)|^2 \, d\xi \lt \infty$ then $\mathcal{G}(g,1,1)$ is not complete and minimal in ${L^2(\mathbb{R})}$. These theorems extend the work of Daubechies and Janssen from the case $(p,q)=(4,4)$. Further refinements and optimal examples are also provided.

A $p$-subgroup $P\neq 1$ of $G$ is called $\mathcal{M}$-permutable in $G$ if there exists a set $\mathcal{ M}_d(P)=\{P_1,\ldots,P_d\}$ of maximal subgroup $P_i$ of $P$ and a subgroup $B$ of $G$ such that: (1) $\bigcap_{i=1}^{d}{P_i}=\Phi(P)$ and $|P : \Phi(P)|=p^d$; (2) $G=PB$ and $P_iB=BP_i \lt G$ for any $P_i$ of $\mathcal{M}_d(P)$. In this paper, we investigate the influence of $\mathcal M$-permutability of Sylow subgroups in finite groups. Some new results about supersolvable groups and formations are obtained.

We consider the dynamical system created by iterating a morphism of a projective variety defined over the field of fractions of a discrete valuation ring. We study the primitive period of a periodic point in this field in relation to the primitive period of the reduced point in the residue field, the order of the action on the cotangent space, and the characteristic of the residue field.

We derive integral representations for $(0,q)$-forms, $q\ge1$, on nonsmooth strictly pseudoconvex domains, the Henkin–Leiterer domains. A $(0,q)$-form, $f$ is written in terms of integral operators acting on $f$, $\bar{\partial} f$, and $\bar{\partial}^{\ast} f$. The representation is applied to derive $L^{\infty}$ estimates.

In this note, we study the foliated structure of a contact metric $(\kappa,\mu)$-space. In particular, using the theory of Legendre foliations, we give a geometric interpretation of the Boeckx's classification of contact metric $(\kappa,\mu)$-spaces and we find necessary conditions for a contact manifold to admit a compatible contact metric $(\kappa,\mu)$-structure. Finally, we prove that any contact metric $(\kappa,\mu)$-space $M$ whose Boeckx invariant $I_M$ is different from $\pm1$ admits a compatible Sasakian or Tanaka–Webster parallel structure according to the circumstance that $|I_M|>1$ or $|I_M| \lt 1$, respectively.

We establish the Poincaré-type inequalities for the composition of the homotopy operator and the projection operator applied to the nonhomogeneous $A$-harmonic equation in John domains. We also obtain some estimates for the integral of the composite operator with a singular density.

A function on the real line is called regulated if it has a left limit and a right limit at each point. If $f$ is a Schwartz distribution on the real line such that $f=F'$ (distributional or weak derivative) for a regulated function $F$ then the regulated primitive integral of $f$ is $\int_{(a,b)}f=F(b-)-F(a+)$, with similar definitions for other types of intervals. The space of integrable distributions is a Banach space and Banach lattice under the Alexiewicz norm. It contains the spaces of Lebesgue and Henstock–Kurzweil integrable functions as continuous embeddings. It is the completion of the space of signed Radon measures in the Alexiewicz norm. Functions of bounded variation form the dual space and the space of multipliers. The integrable distributions are a module over the functions of bounded variation. Properties such as integration by parts, change of variables, Hölder inequality, Taylor’s theorem and convergence theorems are proved.

We show that almost-Einstein hypersurfaces in the Euclidean space are homeomorphic to spheres. The proof relies on universal lower bounds in terms of the Betti numbers for the $L^{n/2}$-norms of the Ricci and traceless Ricci tensor of compact oriented $n$-dimensional hypersurfaces. Certain examples show that the assumption on the codimension is essential.