We characterize the membership in the Schatten ideals $\mathcal {S}_p$, $0<p<\infty $, of composition operators acting on weighted Dirichlet spaces. Our results concern a large class of weights. In particular, we examine the case of perturbed superharmonic weights. Characterization of composition operators acting on weighted Bergman spaces to be in $\mathcal {S}_p$ is also given.

We introduce natural strengthenings of sequential compactness, the r-Ramsey property for each natural number $r\geq 1$ . We prove that metrizable compact spaces are r-Ramsey for all r and give examples of compact spaces that are r-Ramsey but not $(r+1)$ -Ramsey for each $r\geq 1$ (assuming Continuum Hypothesis (CH) for all $r>1$ ). Productivity of the r-Ramsey property is considered.

We establish a theory of noncommutative (NC) functions on a class of von Neumann algebras with a particular direct sum property, e.g., $B({\mathcal H})$ . In contrast to the theory’s origins, we do not rely on appealing to results from the matricial case. We prove that the $k{\mathrm {th}}$ directional derivative of any NC function at a scalar point is a k-linear homogeneous polynomial in its directions. Consequences include the fact that NC functions defined on domains containing scalar points can be uniformly approximated by free polynomials as well as realization formulas for NC functions bounded on particular sets, e.g., the NC polydisk and NC row ball.

Magnitude is a numerical invariant of compact metric spaces, originally inspired by category theory and now known to be related to myriad other geometric quantities. Generalizing earlier results in $\ell _1^n$ and Euclidean space, we prove an upper bound for the magnitude of a convex body in a hypermetric normed space in terms of its Holmes–Thompson intrinsic volumes. As applications of this bound, we give short new proofs of Mahler’s conjecture in the case of a zonoid and Sudakov’s minoration inequality.

In this note, we mainly study operator-theoretic properties on the Besov space $B_{1}$ on the unit disk. This space is the minimal Möbius-invariant space. First, we consider the boundedness of Volterra-type operators. Second, we prove that Volterra-type operators belong to the Deddens algebra of a composition operator. Third, we obtain estimates for the essential norm of Volterra-type operators. Finally, we give a complete characterization of the spectrum of Volterra-type operators.

We prove that $164\, 634\, 913$ is the smallest positive integer that is a sum of two rational sixth powers, but not a sum of two integer sixth powers. If $C_{k}$ is the curve $x^{6} + y^{6} = k$ , we use the existence of morphisms from $C_{k}$ to elliptic curves, together with the Mordell–Weil sieve, to rule out the existence of rational points on $C_{k}$ for various k.

We show that a sufficiently general hypersurface of degree d in $\mathbb {P}^n$ admits a toric Gröbner degeneration after linear change of coordinates if and only if $d\leq 2n-1$.

In the Zermelo–Fraenkel set theory with the Axiom of Choice, a forcing notion is “$\kappa $-distributive” if and only if it is “$\kappa $-sequential.” We show that without the Axiom of Choice, this equivalence fails, even if we include a weak form of the Axiom of Choice, the Principle of Dependent Choice for $\kappa $. Still, the equivalence may still hold along with very strong failures of the Axiom of Choice, assuming the consistency of large cardinal axioms. We also prove that although a $\kappa $-distributive forcing notion may violate Dependent Choice, it must preserve the Axiom of Choice for families of size $\kappa $. On the other hand, a $\kappa $-sequential can violate the Axiom of Choice for countable families. We also provide a condition of “quasiproperness” which is sufficient for the preservation of Dependent Choice, and is also necessary if the forcing notion is sequential.

A classical theorem of Balcar, Pelant, and Simon says that there is a base matrix of height ${\mathfrak h}$, where ${\mathfrak h}$ is the distributivity number of ${\cal P} (\omega ) / {\mathrm {fin}}$. We show that if the continuum ${\mathfrak c}$ is regular, then there is a base matrix of height ${\mathfrak c}$, and that there are base matrices of any regular uncountable height $\leq {\mathfrak c}$ in the Cohen and random models. This answers questions of Fischer, Koelbing, and Wohofsky.

Granville recently asked how the Mahler measure behaves in the context of polynomial dynamics. For a polynomial $f(z)=z^d+\cdots \in {\mathbb C}[z],\ \deg (f)\ge 2,$ we show that the Mahler measure of the iterates $f^n$ grows geometrically fast with the degree $d^n,$ and find the exact base of that exponential growth. This base is expressed via an integral of $\log ^+|z|$ with respect to the invariant measure of the Julia set for the polynomial $f.$ Moreover, we give sharp estimates for such an integral when the Julia set is connected.

This short paper shows a topological obstruction of the existence of certain Lagrangian submanifolds in symplectic $4m$ -manifolds.

Dans cet article on caractérise la lissité du schéma Quot ponctuel emboîté d’une variété lisse—c’est-à-dire l’espace de modules paramétrant les drapeaux de quotients de dimension $0$ d’un faisceau localement libre fixé. Nos résultats étendent la classification de Cheah dans le cadre des schémas de Hilbert ponctuels emboîtés.

In this article, we study the following question asked by Michael Hartz in a recent paper (Every complete Pick space satisfies the column-row property, to appear in Acta Mathematica): which operator spaces satisfy the column–row property? We provide a complete classification of the column–row property (CRP) for noncommutative $L_{p}$-spaces over semifinite von Neumann algebras. We study other relevant properties of operator spaces that are related to the CRP and discuss their existence and nonexistence for various natural examples of operator spaces.

We prove that the Riemann hypothesis is equivalent to the condition $\int _{2}^x\left (\pi (t)-\operatorname {\textrm {li}}(t)\right )\textrm {d}t<0$ for all $x>2$ . Here, $\pi (t)$ is the prime-counting function and $\operatorname {\textrm {li}}(t)$ is the logarithmic integral. This makes explicit a claim of Pintz. Moreover, we prove an analogous result for the Chebyshev function $\theta (t)$ and discuss the extent to which one can make related claims unconditionally.

We characterize zero sets for which every subset remains a zero set too in the Fock space $\mathcal {F}^p$ , $1\leq p<\infty $ . We are also interested in the study of a stability problem for some examples of uniqueness set with zero excess in Fock spaces.

We characterize model polynomials that are cyclic in Dirichlet-type spaces in the unit ball of $\mathbb C^n$, and we give a sufficient capacity condition in order to identify noncyclic vectors.

Let $f:R\to S$ be a ring homomorphism and J be an ideal of S. Then the subring $R\bowtie ^fJ:=\{(r,f(r)+j)\mid r\in R$ and $j\in J\}$ of $R\times S$ is called the amalgamation of R with S along J with respect to f. In this paper, we characterize when $R\bowtie ^fJ$ is a Hilbert ring. As an application, we provide an example of Hilbert ring with maximal ideals of different heights. We also construct non-Noetherian Hilbert rings whose maximal ideals are all finitely generated (unruly Hilbert rings).

We classify braided generalized near-group fusion categories whose global dimension is not an integer; there are exactly two up to Grothendieck equivalence and taking products with braided pointed fusion categories.

Let $r=[a_1(r), a_2(r),\ldots ]$ be the continued fraction expansion of a real number $r\in \mathbb R$ . The growth properties of the products of consecutive partial quotients are tied up with the set admitting improvements to Dirichlet’s theorem. Let $(t_1, \ldots , t_m)\in \mathbb R_+^m$ , and let $\Psi :\mathbb {N}\rightarrow (1,\infty )$ be a function such that $\Psi (n)\to \infty $ as $n\to \infty $ . We calculate the Hausdorff dimension of the set of all $ (x, y)\in [0,1)^2$ such that

$$ \begin{align*} \max\left\{\prod_{i=1}^ma_{n+i}^{t_i}(x), \prod_{i=1}^ma_{n+i}^{t_i}(y)\right\} \geq \Psi(n) \end{align*} $$

$n\geq 1$

Let $X/{\mathbb C}$ be a smooth projective variety. We consider two integral invariants, one of which is the level of the Hodge cohomology algebra $H^*(X,{\mathbb C})$ and the other involving the complexity of the higher Chow groups ${\mathrm {CH}}^*(X,m;{\mathbb Q})$ for $m\geq 0$. We conjecture that these two invariants are the same and accordingly provide some strong evidence in support of this.