The purpose in this paper is to give necessary and sufficient conditions for an LCK structure to be a Vaisman structure on a solvmanifold with a left-invariant complex structure. As an application, we determine a Vaisman solvmanifold such that the commutator of the solvable Lie group is abelian.

The Bergman fan of a matroid is the intersection of tropical hyperplanes defined by the circuits. A tropical basis is a subset of the circuit set that defines the Bergman fan. Yu and Yuster posed a question whether every simple regular matroid has a unique minimal tropical basis of its Bergman fan, and verified it for graphic, cographic matroids and $R_{10}$. We show every simple binary matroid has a unique minimal tropical basis. Since the regular matroid is binary, we positively answered the question.

We show that Kunen’s saturated ideals are not centered. We also evaluate the extent of saturation of Laver’s saturated ideals in terms of -saturation.

We consider the lattice of all compatible topologies on an arbitrary finite-dimensional vector space over a non-discrete valued field whose completion is locally compact. We construct a canonical lattice isomorphism between this lattice and the lattice of all vector subspaces of the vector space whose coefficient field is extended to the complete valued field. Moreover, using this isomorphism, we characterize the continuity of linear maps between such vector spaces, and also characterize compatible topologies that are Hausdorff.

This article is a continuation of [4] in a way. We construct pointed curves on the K3 surfaces treated in [1] which are double covers of rational elliptic surfaces. In some cases we calculate the Weierstrass semigroups of the pointed curves. These pointed curves are the first examples on K3 surfaces that are double covers of rational elliptic surfaces such that we can calculate the Weierstrass semigroups. Moreover, we give bi-elliptic curves of genus 9 or 10 on some such K3 surfaces.

The formal multiple zeta space we consider with a computer is an ${\mathbb{F}}_2$-vector space generated by $2^{k-2}$ formal symbols for a given weight $k$, where the symbols satisfy binary extended double shuffle relations. Up to weight $k=22$, we compute the dimensions of the formal multiple zeta spaces, and verify the dimension conjecture on original extended double shuffle relations of real multiple zeta values. Our computations adopt Gaussian forward elimination and give information for spaces filtered by depth. We can observe that the dimensions of the depth-graded formal multiple zeta spaces have a Pascal triangle pattern expected by the Hoffman mult-indices.

Let $f$, $g$ be fixed coprime positive integers with . Recently, T. Miyazaki and N. Terai [11] conjectured that the equation $f^x+{(f+g)}^y=g^z$ has no positive integer solutions $(x,y,z)$, except for certain known pairs $(f,g)$. This is a problem that is far from being solved. Let $r$ be an odd positive integer with . In this paper, using Baker’s method with some known results on the generalized Lebesgue-Nagell equations, we prove that if $f=2^r$ and one of the following conditions is satisfied, then the above conjecture is true. (i) Either $g$ or $f+g$ has a divisor $d$ with $d\equiv 5$ or $7$ $(\mathrm{mod}8)$. (ii) or $167748\log g$ according to $g\equiv 1$ or $3$ $(\mathrm{mod}8)$.