A quasi-complete intersection (q.c.i.) ideal of a local ring is an ideal with “free exterior Koszul homology”; the definition can also be understood in terms of vanishing of André-Quillen homology functors. Principal q.c.i. ideals are well understood, but few constructions are known to produce q.c.i. ideals of grade zero that are not principal. This paper examines the structure of q.c.i. ideals. We exhibit conditions on a ring $R$ which guarantee that every q.c.i. ideal of $R$ is principal. On the other hand, we give an example of a minimal q.c.i. ideal $I$ which does not contain any principal q.c.i. ideals and is not embedded, in the sense that no faithfully flat extension of $I$ can be written as a quotient of complete intersection ideals. We also describe a generic situation in which the maximal ideal of $R$ is an embedded q.c.i. ideal that does not contain any principal q.c.i. ideals.

In this paper, we study the Dirichlet problem for the minimal surface equation in $\mathrm{Sol}_{3}$ with possible infinite boundary data, where $\mathrm{Sol}_{3}$ is the non-Abelian solvable $3$-dimensional Lie group equipped with its usual left-invariant metric that makes it into a model space for one of the eight Thurston geometries. Our main result is a Jenkins–Serrin type theorem which establishes necessary and sufficient conditions for the existence and uniqueness of certain minimal Killing graphs with a non-unitary Killing vector field in $\mathrm{Sol}_{3}$.

The notion of thickness, introduced in (Math. Ann. 344 (2009) 543–595), is one of the first tools developed to study the quasi-isometric behavior of weakly relatively hyperbolic groups. In this paper, we further this exploration through a relationship between thickness and the divergence of geodesics. We construct examples, for every positive integer $n$, of $\operatorname{CAT}(0)$ groups which are thick of order $n$ and with polynomial divergence of order $n+1$. With respect to thickness, these examples show the non-triviality at each level of the thickness hierarchy defined in (Math. Ann. 344 (2009) 543–595). With respect to divergence, our examples provide an answer to questions of Gromov (In Geometric Group Theory (1993) 1–295 Cambridge Univ. Press) and Gersten (Geom. Funct. Anal. 4 (1994) 633–647; Geom. Funct. Anal. 4 (1994) 37–51). The divergence questions were independently answered by Macura in ($\operatorname{CAT}(0)$ spaces with polynomial divergence of geodesics (2011) Preprint).

We also provide tools for obtaining both lower and upper bounds on the divergence of geodesics and spaces, and we prove an effective quadratic lower bound for Morse quasi-geodesics in $\operatorname{CAT}(0)$ spaces, generalizing results of Kapovich–Leeb and Bestvina–Fujiwara (Geom. Funct. Anal. 8 (1998) 841–852; Geom. Funct. Anal. 19 (2009) 11–40).

In the final section, we obtain linear and quadratic bounds on the length of the shortest conjugators for various families of groups. For general $3$-manifold groups, sharp estimates are provided. We also consider mapping class groups, where we provide a new streamlined proof of the length of shortest conjugators which contains the corresponding results of Masur–Minsky in the pseudo-Anosov case (Geom. Funct. Anal. 10 (2000) 902–974) and Tao in the reducible case (Geom. Funct. Anal. 23 (2013) 415–466).

In this paper, it is shown that a polyomino is balanced if and only if it is simple. As a consequence, one obtains that the coordinate ring of a simple polyomino is a normal Cohen–Macaulay domain.

Billey and Braden defined maps on flag manifolds that are the geometric counterpart of permutation patterns. A section of their pattern map is an embedding of the flag manifold of a Levi subgroup into the full flag manifold. We give two expressions for the induced map on cohomology. One is in terms of generators and the other is in terms of the Schubert basis. We show that the coefficients in the second expression are naturally Schubert structure constants and therefore positive. Similar results hold for $K$-theory, generalizing known formulas in type $A$ for cohomology and $K$-theory.

It is known that hyperbolic spaces have strict negative type, a condition on the distances of any finite subset of points. We show that they have strong negative type, a condition on every probability distribution of points (with integrable distance to a fixed point). This implies that the function of expected distances to points determines the probability measure uniquely. It also implies that the distance covariance test for stochastic independence, introduced by Székely, Rizzo and Bakirov, is consistent against all alternatives in hyperbolic spaces. We prove this by showing an analogue of the Cramér–Wold device.

In this paper, we will give Schwarz–Pick type estimates of arbitrary order partial derivatives for bounded pluriharmonic mappings defined in the unit polydisk. Our main results are generalizations of results of Colonna for planar harmonic mappings in (Indiana Univ. Math. J. 38 (1989) 829–840).

We prove some weighted estimates for two kinds of Littlewood–Paley operators related to the Riesz potentials, which can be used to characterize the weighted Sobolev spaces. Also, we show the boundedness from the weighted Hardy space $H^{1}_{w}$ to the weighted weak $L^{1}$ space of a Littlewood–Paley operator arising from spherical means.

We describe coefficient ideals for both $(x,y)$-primary monomial ideals in $k[x,y]$ and $\mathfrak{m}$-primary ideals in two-dimensio-nal regular local rings $(R,\mathfrak{m})$ by linking them to certain ideals of reduction number one. In the monomial case, we then explicitly determine the generators of a coefficient ideal by showing their symmetric relationship to the generators of the associated reduction number one ideal.

We prove that every (not necessarily linear nor continuous) 2-local triple derivation on a von Neumann algebra $M$ is a triple derivation, equivalently, the set $\operatorname{Der}_{t}(M)$, of all triple derivations on $M$, is algebraically 2-reflexive in the set $\mathcal{M}(M)=M^{M}$ of all mappings from $M$ into $M$.

In this paper, we introduce p-mapping spaces for p-operator spaces on $L_{p}$ spaces, which can be regarded as p-generalization of mapping spaces for operator spaces. We then apply p-mapping spaces to study the p-local reflexivity for p-operator spaces on $L_{p}$ spaces.

The duality of uniform approximation property for Banach spaces is well known. In this note, we establish, under the assumption of local reflexivity, the duality of uniform approximation property in the category of operator spaces.

Let $S$ be an operator system in $B(H)$ and let $A$ be its generated $C^{*}$-algebra. We give a new characterization of Arveson’s unique extension property for unital completely positive maps on $S$. We also show that when $A$ is a Type I $C^{\ast}$-algebra, if every irreducible representation of $A$ is a boundary representation for $S$, then every unital completely positive map on $A$ with codomain $A"$ that fixes $S$ also fixes $A$.

We prove that a general (not necessarily symmetric) Lévy process killed on exiting a bounded open set (without regular condition on the boundary) is intrinsically ultracontractive, provided that $B(0,R_{0})\subseteq\operatorname{supp} (\nu)$ for some constant $R_{0}>0$, where $\operatorname{supp} (\nu)$ denotes the support of the associated Lévy measure $\nu$. For a symmetric Lévy process killed on exiting a bounded Hölder domain of order $0$, we also obtain the intrinsic ultracontractivity under much weaker assumption on the associated Lévy measure.