High order and fractional PDEs have become prominent in theory and in modeling many phenomena. Here, we focus on the regularizing effect of a large class of memoryful high-order or time-fractional PDEs—through their fundamental solutions—on stochastic integral equations (SIEs) driven by space–time white noise. Surprisingly, we show that maximum spatial regularity is achieved in the fourth-order-bi-Laplacian case; and any further increase in the spatial-Laplacian order is entirely translated into additional temporal regularization of the SIE. We started this program in [Discrete Contin. Dyn. Syst. Ser. A 33 (2013) 413–463, Stoch. Dyn. 6 (2006) 521–534], where we introduced two different stochastic versions of the fourth order memoryful PDE associated with the Brownian-time Brownian motion (BTBM): (1) the BTBM SIE and (2) the BTBM SPDE, both driven by space–time white noise. Under wide conditions, we showed the existence of random field locally-Hölder solutions to the BTBM SIE with striking and unprecedented time-space Hölder exponents, in spatial dimensions $d=1,2,3$. In particular, we proved that the spatial regularity of such solutions is nearly locally Lipschitz in $d=1,2$. This gave, for the first time, an example of a space–time white noise driven equation whose solutions are smoother than the corresponding Brownian sheet in either time or space.

In this paper, we introduce the $2\beta^{-1}$-order $\beta$-inverse-stable-Lévy-time Brownian motion ($\beta$-ISLTBM) SIEs, $\beta\in \{1/2^{k};k\in\mathbb{N}\}$, driven by space–time white noise. Based on the dramatic regularizing effect of the BTBM density ($\beta=1/2$), and since the kernels in these $\beta$-ISLTBM SIEs are fundamental solutions to higher order Laplacian PDEs; one may suspect that we get even more dramatic spatial regularity than the BTBM SIE case. We show, however, that the BTBM SIE spatial regularity and its random field third spatial dimension limit are maximal among all $\beta$-ISLTBM SIEs—no matter how high we take the order $1/\beta$ of the Laplacian. This gives a limit as to how far we can push the SIEs spatial regularity when driven by the rough white noise. Furthermore, we show that increasing the order of the Laplacian $\beta^{-1}$ beyond the BTBM bi-Laplacian manifests entirely as increased temporal regularity of our random field solutions that asymptotically approaches that of the Brownian sheet as $\beta\searrow0$. Our solutions are both direct and lattice limit solutions. We treat many stochastic fractional PDEs and their corresponding higher order SPDEs, including BTBM and $\beta$-inverse-stable-Lévy-time Brownian motion SPDEs, in separate articles.

We study mappings of finite distortion whose distortion functions are locally subexponentially integrable. We establish a local modulus of continuity estimate for the inverse of such a map. As applications, we describe the possible expansion and compression of certain Hausdorff measures and Minkowski contents under such mappings. We also exhibit examples that describe the extent to which our results are sharp.

We construct an infinite family of real quadratic number fields with class group of $2$-$\operatorname{rank} =3$, $4$-$\operatorname{rank} =1$ and finite Hilbert $2$-class field tower.

Let $S^{n}(X)$ be the $n$-fold symmetric product of a compact connected Riemann surface $X$ of genus $g$ and gonality $d$. We prove that $S^{n}(X)$ admits a Kähler structure such that all the holomorphic bisectional curvatures are nonpositive if and only if $n<d$. Let $\mathcal{Q}_{X}(r,n)$ be the Quot scheme parametrizing the torsion quotients of $\mathcal{O}^{\oplus r}_{X}$ of degree $n$. If $g\geq 2$ and $n\leq 2g-2$, we prove that $\mathcal{Q}_{X}(r,n)$ does not admit a Kähler structure such that all the holomorphic bisectional curvatures are nonnegative.

Our aim in this paper is to deal with Lindelöf type theorems for monotone Sobolev functions in Orlicz spaces.

A piecewise smooth domain is said to have generic corners if the corners are generic CR manifolds. It is shown that a biholomorphic mapping from a piecewise smooth pseudoconvex domain with generic corners in complex Euclidean space that satisfies Condition R to another domain extends as a smooth diffeomorphism of the respective closures if and only if the target domain is also piecewise smooth with generic corners and satisfies Condition R. Further it is shown that a proper map from a domain with generic corners satisfying Condition R to a product domain of the same dimension extends continuously to the closure of the source domain in such a way that the extension is smooth on the smooth part of the boundary. In particular, the existence of such a proper mapping forces the smooth part of the boundary of the source to be Levi degenerate.

Motivated by connections between dimension groups and good measures or minimal actions on Cantor sets (especially Töplitz), we find realizations of classes of dimension groups as limits of primitive matrices all of which have equal column sums, or equal row sums.

Let $H=\mathrm{O}(n)\times\mathrm{O}(1)$ be an anisotropic subgroup of $G=\mathrm{O} (n,1)$ and let $\mathbb{A} $ be the adele ring of $k=\mathbb{Q}$. Consider the periods

\[(E_{\varphi },F)_{H}=\int_{H_{k}\backslash H_\mathbb{A}}E_{\varphi}\cdot{\overline {F}},\]

of an Eisenstein series $E_{\varphi}$ on $G$ against a form $F$ on $H$. Relying on a variant of Levi–Sobolev spaces, we describe certain Poincaré series as fundamental solutions for the Laplacian, and use them to establish a spectral identity concerning the second moments (in $F$-aspect) of $E_{\varphi }$.

The category of locally compact quantum groups can be described as either Hopf $*$-homomorphisms between universal quantum groups, or as bicharacters on reduced quantum groups. We show how Sołtan’s quantum Bohr compactification can be used to construct a “compactification” in this category. Depending on the viewpoint, different C$^{*}$-algebraic compact quantum groups are produced, but the underlying Hopf $*$-algebras are always, canonically, the same. We show that a complicated range of behaviours, with C$^{*}$-completions between the reduced and universal level, can occur even in the cocommutative case, thus answering a question of Sołtan. We also study such compactifications from the perspective of (almost) periodic functions. We give a definition of a periodic element in $L^{\infty}(\mathbb{G})$, involving the antipode, which allows one to compute the Hopf $*$-algebra of the compactification of $\mathbb{G} $; we later study when the antipode assumption can be dropped. In the cocommutative case, we make a detailed study of Runde’s notion of a completely almost periodic functional—with a slight strengthening, we show that for [SIN] groups this does recover the Bohr compactification of $\hat{G}$.

For each $n\in \mathbb{N} $ a Banach space $\mathfrak{X}_{_{^{0,1}}}^{n}$ is constructed having the property that every normalized weakly null sequence generates either a $c_{0}$ or $\ell_{1}$ spreading models and every subspace has weakly null sequences generating both $c_{0}$ and $\ell_{1}$ spreading models. The space $\mathfrak{X}_{_{^{0,1}}}^{n}$ is also quasiminimal and for every infinite dimensional closed subspace $Y$ of $\mathfrak{X}_{_{^{0,1}}}^{n}$, for every $S_{1},S_{2},\ldots,S_{n+1}$ strictly singular operators on $Y$, the operator $S_{1}S_{2}\cdots S_{n+1}$ is compact. Moreover, for every subspace $Y$ as above, there exist $S_{1},S_{2},\ldots,S_{n}$ strictly singular operators on $Y$, such that the operator $S_{1}S_{2}\cdots S_{n}$ is non-compact.