We show that the iterative logarithm of each non-linear entire function is differentially transcendental over the ring of entire functions, and we give a sufficient criterion for such an iterative logarithm to be differentially transcendental over the ring of convergent power series. Our results apply, in particular, to the exponential generating function of a sequence arising from work of Shadrin and Zvonkine on Hurwitz numbers.

This paper completely characterizes the commuting problem of two small Hankel operators acting on the harmonic Bergman space with the symbols one being bounded and another being quasihomogeneous, or both being harmonic. The characterizations for semi-commuting problem and the product of two small Hankel operators being another small Hankel operator for certain class of symbols are also obtained.

Given a principally polarized Abelian variety $(A,\Theta)$, we give a characterization of all elliptic curves that lie on $A$ in terms of intersection numbers of divisor classes in its Néron–Severi group.

We construct a Schauder basis for $L_{1}$ consisting of non-negative functions and investigate unconditionally basic and quasibasic sequences of non-negative functions in $L_{p}$, $1\le p<\infty$.

The Cremona group $\operatorname{Bir}(\mathbb{P}^{2}_{\mathbb{C}})$ is the group of birational self-maps of $\mathbb{P}^{2}_{\mathbb{C}}$. Using the action of $\operatorname{Bir}(\mathbb{P}^{2}_{\mathbb{C}})$ on the Picard-Manin space of $\mathbb{P}^{2}_{\mathbb{C}}$, we characterize its solvable subgroups. If $\mathrm{G}\subset\operatorname{Bir}(\mathbb{P}^{2}_{\mathbb{C}})$ is solvable, nonvirtually Abelian, and infinite, then up to finite index: either any element of $\mathrm{G}$ is of finite order or conjugate to an automorphism of $\mathbb{P}^{2}_{\mathbb{C}}$, or $\mathrm{G}$ preserves a unique fibration that is rational or elliptic, or $\mathrm{G}$ is, up to conjugacy, a subgroup of the group generated by one hyperbolic monomial map and the diagonal automorphisms.

We also give some corollaries.

We discuss an alternate method for computing the Szlenk index of an arbitrary $w^{*}$ compact subset of the dual of a Banach space. We discuss consequences of this method as well as offer simple, alternative proofs of a number of results already found in the literature.

It is known that the polyomino ideal arising from a simple polyomino comes from a finite bipartite graph and, in particular, it is a prime ideal. A class of nonsimple polyominoes $\mathcal{P}$ for which the polyomino ideal $I_{\mathcal{P}}$ is a prime ideal and for which $I_{\mathcal{P}}$ cannot come from a finite simple graph will be presented.

This paper contains some further applications of the study of knot diagrams by genus. Introducing a procedure of regularization for knot generators, and using invariants derived from the Jones polynomial (degrees, congruences, and the Fiedler–Polyak–Viro Gauß diagram formulas for its Vassiliev invariants), we examine the existence of genus-minimizing diagrams for almost alternating and almost positive knots. In particular, we examine the existence of such knots such that either all or none of their almost alternating/positive diagrams have the minimal genus property. We prove that the genus of almost positive non-split links is determined by the Alexander polynomial.

Given an inclusion $\mathcal{D}\subseteq\mathcal{C}$ of unital $C^{*}$-algebras (with common unit), a unital completely positive linear map $\Phi$ of $\mathcal{C}$ into the injective envelope $I(\mathcal{D})$ of $\mathcal{D}$ which extends the inclusion of $\mathcal{D}$ into $I(\mathcal{D})$ is a pseudo-expectation. Pseudo-expectations are generalizations of conditional expectations, but with the advantage that they always exist. The set $\operatorname{PsExp}(\mathcal{C},\mathcal{D})$ of all pseudo-expectations is a convex set, and when $\mathcal{D}$ is Abelian, we prove a Krein–Milman type theorem showing that $\operatorname{PsExp}(\mathcal{C},\mathcal{D})$ can be recovered from its set of extreme points. In general, $\operatorname{PsExp}(\mathcal{C},\mathcal{D})$ is not a singleton. However, there are large and natural classes of inclusions (e.g., when $\mathcal{D}$ is a regular MASA in $\mathcal{C}$) such that there is a unique pseudo-expectation. Uniqueness of the pseudo-expectation typically implies interesting structural properties for the inclusion. For general inclusions of $C^{*}$-algebras with $\mathcal{D}$ Abelian, we give a characterization of the unique pseudo-expectation property in terms of order structure; and when $\mathcal{C}$ is Abelian, we are able to give a topological description of the unique pseudo-expectation property.

As applications, we show that if an inclusion $\mathcal{D}\subseteq\mathcal{C}$ has a unique pseudo-expectation $\Phi$ which is also faithful, then the$C^{*}$-envelope of any operator space $\mathcal{X}$ with $\mathcal{D}\subseteq\mathcal{X}\subseteq\mathcal{C}$ is the $C^{*}$-subalgebra of $\mathcal{C}$ generated by $\mathcal{X}$; we also show that for many interesting classes of $C^{*}$-inclusions, having a faithful unique pseudo-expectation implies that $\mathcal{D}$ norms $\mathcal{C}$, although this is not true in general.

We consider the martingale problem for superprocess with interactive immigration mechanism. The uniqueness of the solution to this martingale problem is established using the strong uniqueness of the solution to a corresponding SPDE, which is obtained by an extended version of the Yamada–Watanabe argument.

Related to a conjecture of Tom Wolff, we solve a singular Neumann problem for a linearized $p$-Laplace equation in the unit disk.