Let R be a Gorenstein artin algebra, and let U be a fixed left R-module and k ⩾ 0. When U is Gorenstein injective, if the U-codominant dimension of any injective left R-module is at most k + 1, then the U-dominant dimension of any projective left R-module is at most k + 1. Dually, when U is Gorenstein projective, if the U-dominant dimension of any projective left R-module is at most k + 1, then the U-codominant dimension of any injective left R-module is at most k + 1.

Let R and S be rings and _RC_S a semidualizing bimodule, and let be a subcategory of the Auslander class and . Then for any left R-module M, the -projective dimension of Hom_R (C,M) is at most the -projective dimension of M, and they are identical when M is in the Bass class . If _RC_S is faithful and is resolving, then in a short exact sequence of left R-modules, the -projective dimensions of any two terms can determine an upper bound of that of the third term. Furthermore, we apply these results to the cases of being the subcategories of (weak) flat modules, projective modules and respectively. Some known results are obtained as corollaries.

Let k be an even integer and f ∈ S_k(Γ₀(N)) be a newform of weight k. We prove that for each integer N ≥ 16 and for k ≥ k_N,ε (depending on N), all of nonzero zeros of the odd period polynomial associated to the newform f are on the circle |z| = 1 / √N. For each integer 3 ≤ N ≤ 15, we also investigate the location of the zeros of the odd period polynomial associated to the newform f.

Kodaira-Thurston manifold is a compact nilmanifold, and it has a Vaisman structure. The purpose in this paper is to calculate the sectional curvature and the Ricci tensor on Kodaira-Thurston manifold. Moreover, we consider curvatures and the Ricci tensor on a compact Vaisman solvmanifold.

We consider certain local-global principles related with the isotropy and some splitting problems for connected linear algebraic groups over global fields. The main tools are certain reciprocity law for Tits indices of almost simple groups due to Prasad and Rapinchuk, Harder's Hasse principle for homogeneous projective spaces of reductive groups for number fields and their extensions to global function fields.

In this paper, we consider translators (for the mean curvature flow) given by a graph of a function on a symmetric space G/K of compact type which is invariant under a hyperpolar action on G/K. First, in the case of G/K = SO(n + 1)/SO(n), SU(n + 1)/S(U(1) × U(n)), Sp(n + 1)/(Sp(1) × Sp(n)) or F₄/Spin(9), we classify the shapes of translators in G/K × given by the graphs of functions on G/K which are invariant under the isotropy action K ↷ G/K. Next, in the case where G/K is of higher rank, we investigate translators in G/K × given by the graphs of functions on G/K which are invariant under a hyperpolar action H ↷ G/K of cohomogeneity two.

We use the BFK-gluing formula for zeta-determinants to compute the zeta-determinant and analytic torsion of a metric mapping torus induced from an isometry. As applications, we compute the zeta-determinants of the Laplacians defined on a Klein bottle K and some compact co-Kähler manifold T_φ. We also show that a metric mapping torus and a Riemannian product manifold with a round circle have the same heat trace asymptotic expansions. We finally compute the analytic torsion of a metric mapping torus for the Witten deformed Laplacian and recover the result of J. Marcsik in [16].