Toeplitz CAR flows are a class of $E_0$-semigroups including the first type III example constructed by R. T. Powers. We show that the Toeplitz CAR flows contain uncountably many mutually non-cocycle-conjugate $E_0$-semigroups of type III. We also generalize the type III criterion for Toeplitz canonical anticommutation relation (CAR) flows employed by Powers (and later refined by W. Arveson), and show that Toeplitz CAR flows are always either of type I or type III.

This new concept of continued fractional measure of irrationality for the real number $a$ is introduced with the help of the classical measure of irrationality. Some relationships between this new and the classical measures are included.

Let $B$ be a Banach algebra with bounded approximate identity, and let $M\left(B\right)$ be its multiplier algebra. If there exists a continuous linear injection $B^{*}\to M\left(B\right)$ such that, for every $b\in B$ and every $u,v\in B^{*}$, $\langle u,vb{\rangle }_B=\langle v,bu{\rangle }_B$, then $M\left(B\right)$ is a dual Banach algebra and the following are equivalent:

(i) $B$ is amenable;

(ii) $M\left(B\right)$ is Connes amenable;

(iii) $M\left(B\right)$ has a normal, virtual diagonal.

In this article we study coverings with prescribed ramification from the point of view of the Sato Grassmannian and of the algebro-geometric theory of solitons. We show that the moduli space of such coverings, which is a Hurwitz scheme, is a subscheme of the Grassmannian. We give its equations and show that there is a Virasoro group that uniformizes it. We also characterize when a curve is a covering in terms of bilinear identities.

In this article we prove that if Deligne’s conjecture holds for motives associated to Hilbert modular forms of weight at least 3, then Deligne’s conjecture holds for arbitrary base change to totally real number fields of motives associated to Hilbert modular forms of weight at least 3.

This article addresses frame multiresolution analyses (FMRAs) and FMRA frame wavelets in the setting of reducing subspaces of $L^2\left({\mathbb{R}}^d\right)$. For a general expansive matrix, we obtain a characterization and some conditions for a frame-scaling function to generate an FMRA, and we prove that an arbitrary reducing subspace must admit an FMRA. For an expansive matrix $M$ with $|\det M|=2$, we establish a sufficient and necessary condition for FMRAs to admit a single FMRA frame wavelet, give an explicit construction of FMRA frame wavelets, and study the relation between $s$-frame wavelets and FMRA frame wavelets. These results are also new in the setting of $L^2\left({\mathbb{R}}^d\right)$.

A Schrödinger equation with a merging pair of a simple pole and a simple turning point (called MPPT equation for short) is studied from the viewpoint of exact Wentzel-Kramers-Brillouin (WKB) analysis. In a way parallel to the case of merging-turning-points (MTP) equations, we construct a WKB-theoretic transformation that brings an MPPT equation to its canonical form (the $\infty$-Whittaker equation in this case). Combining this transformation with the explicit description of the Voros coefficient for the Whittaker equation in terms of the Bernoulli numbers found by Koike, we discuss analytic properties of Borel-transformed WKB solutions of an MPPT equation.

We prove a Gauss-Bonnet-type formula for Riemann-Finsler surfaces of nonconstant indicatrix volume and with regular piecewise $C^{\infty }$-boundary. We give a Hadamard-type theorem for $N$-parallels of a Landsberg surface.

For a compact Riemann surface, $(n+1)$-tuple $x:=(x_0,\dots ,x_n)$ of points on it, and a holomorphic vector bundle with an integrable connection on the open Riemann surface $X_x$ deprived of $(n+1)$ points $x_0,\dots ,x_n$, let $\mathcal{L}$ be the local system of horizontal sections of the connection. In this article, we give a suitable covering of $X_x$ to calculate the Čech cohomology and describe the isomorphism between the cohomology and the twisted de Rham cohomology, which is the cohomology of the complex with the differentials given by the connection. This isomorphism is given by the integrations over Aomoto’s regularized paths, the so-called Euler type integrals.

For the family $\{X_x{\}}_x$ parametrized by $x$, we give a variant of the isomorphism.

In this article, we study the local dynamical structure of a rational mapping $F$ of ${\mathbf{P}}^2$ at a fixed indeterminate point $p$. Using a sequence of blowups, we construct a family $\{{\tilde{W}}_{\mathbf{j}}{\}}_{\mathbf{j}\in J}$ of germs of holomorphic curve at the point $p$, where $J$ is a subset of a Cantor set $\{1,2{\}}^{\mathbf{N}}$. This is a new construction for a Cantor bouquet.