We give a classification of all nonsymplectic automorphisms of prime order $p$ acting on irreducible holomorphic symplectic four-folds deformation equivalent to the Hilbert scheme of two points on a K3 surface, for $p=2,3$, and $7\le p\le 19$. Our classification relates some invariants of the fixed locus to the isometry classes of two natural lattices associated to the action of the automorphism on the second cohomology group with integer coefficients. In several cases we provide explicit examples. As an application, we find new examples of nonnatural nonsymplectic automorphisms of order $3$.

We construct new families of nonhyperelliptic Lefschetz fibrations by applying the daisy substitutions to the families of words $(c_1{c}_2\cdots c_{2g-1}{c}_{2g}{c_{2g+1}}^2{c}_{2g}\cdot c_{2g-1}\cdots c_2{c}_1{)}^2=1$, $(c_1{c}_2\cdots c_{2g}{c}_{2g+1}{)}^{2g+2}=1$, and $(c_1{c}_2\cdots c_{2g-1}{c}_{2g}{)}^{2(2g+1)}=1$ in the mapping class group ${\Gamma }_g$ of the closed orientable surface of genus $g$, and we study the sections of these Lefschetz fibrations. Furthermore, we show that the total spaces of some of these Lefschetz fibrations are irreducible exotic $4$-manifolds, and we compute their Seiberg–Witten invariants. By applying the knot surgery to the family of Lefschetz fibrations obtained from the word $(c_1{c}_2\cdots c_{2g}{c}_{2g+1}{)}^{2g+2}=1$ via daisy substitutions, we also construct an infinite family of pairwise nondiffeomorphic irreducible symplectic and nonsymplectic $4$-manifolds homeomorphic to $(g^2-g+1){\mathbb{CP}}^2\#(3g^2-g(k-3)+2k+3){\overline{\mathbb{CP}}}^2$ for any $g\ge 3$ and $k=2,\dots ,g+1$.

We prove that every sectional-hyperbolic Lyapunov stable set contains a nontrivial homoclinic class.

This article focuses on deformations of hyperbolic cone structures under the assumption that the length of the singularity remains uniformly bounded during the deformation. Let $M$ be a closed, orientable, and irreducible $3$-manifold, and let $\Sigma$ be an embedded link in $M$. For a collapsing sequence of hyperbolic cone structures with topological type $(M,\Sigma )$ and with uniformly bounded lengths of singularities, we prove that $M$ is either Seifert fibered or a $\mathrm{Sol}$ manifold.

In this article, we study the continuity of some commutators generated by the Calderón–Zygmund singular integral operator, the fractional integral operator, the Marcinkiewicz integral operator, and Lipschitz functions on Herz-type Hardy spaces with variable exponent.

This article gives an explicit formula for the Ehrhart quasipolynomial of certain 2-dimensional polyhedra in terms of invariants of surface quotient singularities. Also, a formula for the dimension of the space of quasihomogeneous polynomials of a given degree is derived. This admits an interpretation as a numerical adjunction formula for singular curves on the weighted projective plane.

We prove that, for any type ${\mathrm{III}}_1$ free product factor, its continuous core is full if and only if its $\tau$-invariant is the usual topology on the real line. This trivially implies, as a particular case, the same result for free Araki–Woods factors. Moreover, our method shows the same result for full (generalized) Bernoulli crossed product factors of type ${\mathrm{III}}_1$.

We suggest an algorithm computing, in some cases, an explicit generating set for the Néron–Severi lattice of a Delsarte surface. In a few special cases, including those of Fermat surfaces and cyclic Delsarte surfaces that were previously conjectured in the literature, we show that certain “obvious” divisors do generate the lattice. The proof is based on the computation of the Alexander module related to a certain abelian covering.

In this article we are concerned with Trudinger’s inequality and continuity for Riesz potentials of functions in grand Musielak–Orlicz–Morrey spaces over nondoubling metric measure spaces.

With the help of the maximal function characterizations of Besov-type spaces with variable smoothness and integrability we prove the characterization by ball means of differences for these function spaces.

We determine the $T$-equivariant integral cohomology ring of the flag manifold $Sp\left(n\right)/T$ as a quotient ring of a polynomial ring, where $T$ is a maximal torus of $Sp\left(n\right)$ and acts on $Sp\left(n\right)/T$ by left multiplication.

Following an approach of Ozawa, we show that several semidirect products are not weakly amenable. As a consequence, we are able to completely characterize the simply connected Lie groups that are weakly amenable.