In this paper, we prove that, for an integer $r$ with $(r,6)=1$ and $0<r<24$ and a nonnegative even integer $s$, the set $$\left\{\eta \right(24\tau {)}^{r}f\left(24\tau \right):f\left(\tau \right)\in M_s\left(1\right)\}$$ is isomorphic to $$S_{r+2s-1}^{\mathrm{new}}(6,-(\frac{8}{r}),-(\frac{12}{r}\left)\right)\otimes \left(\frac{12}{\cdot }\right)$$ as Hecke modules under the Shimura correspondence. Here $M_s\left(1\right)$ denotes the space of modular forms of weight $s$ on ${\Gamma }_0\left(1\right)=SL(2,\mathbb{Z})$, $S_{2k}^{\mathrm{new}}(6,{ϵ}_2,{ϵ}_3)$ is the space of newforms of weight $2k$ on ${\Gamma }_0\left(6\right)$ that are eigenfunctions with eigenvalues ${ϵ}_2$ and ${ϵ}_3$ for Atkin–Lehner involutions $W_2$ and $W_3$, respectively, and the notation $\otimes (12/\cdot )$ means the twist by the quadratic character $(12/\cdot )$. There is also an analogous result for the cases $(r,6)=3$.

In this paper, coupled systems $$u_t+u_{xxx}+P(u,v{)}_x=0,$$ $$v_t+v_{xxx}+Q(u,v{)}_x=0$$ of Korteweg–de Vries type are considered, where $u=u(x,t)$, $v=v(x,t)$ are real-valued functions and where $x,t\in \mathbb{R}$. Here, subscripts connote partial differentiation and $$P(u,v)=A{u}^2+Buv+C{v}^2\text{and}Q(u,v)=D{u}^2+Euv+F{v}^2$$ are quadratic polynomials in the variables $u$ and $v$. Attention is given to the pure initial-value problem in which $u(x,t)$ and $v(x,t)$ are both specified at $t=0$, namely, $$u(x,0)=u_0\left(x\right)\text{and}v(x,0)=v_0\left(x\right),$$ for $x\in \mathbb{R}$. Under suitable conditions on $P$ and $Q$, global well-posedness of this problem is established for initial data in the $L^2$-based Sobolev spaces $H^s\left(\mathbb{R}\right)\times H^s\left(\mathbb{R}\right)$ for any $s>-3/4$.

We investigate operator-valued monotone independence, a noncommutative version of independence for conditional expectation. First we introduce operator-valued monotone cumulants to clarify the whole theory and show the moment-cumulant formula. As an application, one can obtain an easy proof of the central limit theorem for the operator-valued case. Moreover, we prove a generalization of Muraki’s formula for the sum of independent random variables and a relation between generating functions of moments and cumulants.

Given a local ring containing a field, we define and investigate a family of invariants that includes the Lyubeznik numbers but captures finer information. These generalized Lyubeznik numbers are defined in terms of $D$-modules and are proved well defined using a generalization of the classical version of Kashiwara’s equivalence for smooth varieties; we also give a definition for finitely generated $K$-algebras. These new invariants are indicators of $F$-singularities in characteristic $p>0$ and have close connections with characteristic cycle multiplicities in characteristic zero. We characterize the generalized Lyubeznik numbers associated to monomial ideals and compute examples of those associated to determinantal ideals.

We continue our study of the relation between log minimal models and various types of Zariski decompositions. Let $(X,B)$ be a projective log canonical pair. We will show that $(X,B)$ has a log minimal model if either $K_X+B$ birationally has a Nakayama–Zariski decomposition with nef positive part, or if $K_X+B$ is big and birationally has a Fujita–Zariski or Cutkosky–Kawamata–Moriwaki–Zariski decomposition. Along the way we introduce polarized pairs $(X,B+P)$, where $(X,B)$ is a usual projective pair and where $P$ is nef, and we study the birational geometry of such pairs.

This article presents two constructions motivated by a conjecture of van den Dries and Miller concerning the restricted analytic field with exponentiation. The first construction provides an example of two o-minimal expansions of a real closed field that possess the same field of germs at infinity of one-variable functions and yet define different global one-variable functions. The second construction gives an example of a family of infinitely many distinct maximal polynomially bounded reducts (all this in the sense of definability) of the restricted analytic field with exponentiation.