LOCALIZATIONS OF THE HEARTS OF COTORSION PAIRS

In this article, we study localizations of hearts of cotorsion pairs ($\mathcal{U}, \mathcal{V}$) where $\mathcal{U}$ is rigid on an extriangulated category $\mathcal{B}$ . The hearts of such cotorsion pairs are equivalent to the functor categories over the stable category of $\mathcal{U}$ ( $\bmod \underline{\mathcal{U}}$ ). Inspired by Marsh and Palu (Nagoya Math. J.225(2017), 64–99), we consider the mutation (in the sense of Iyama and Yoshino, Invent. Math.172(1) (2008), 117–168) of $\mathcal{U}$ that induces a cotorsion pair ( $\mathcal{U}^{\prime}, \mathcal{V}^{\prime}$ ). Generally speaking, the hearts of ( $\mathcal{U}, \mathcal{V}$ ) and ( $\mathcal{U}^{\prime}, \mathcal{V}^{\prime}$ ) are not equivalent to each other, but we will give a generalized pseudo-Morita equivalence between certain localizations of their hearts.

Explicit logarithmic formulas of special values of hypergeometric functions 3F2

In [M. Asakura, N. Otsubo and T. Terasoma, An algebro-geometric study of special values of hypergeometric functions [Formula: see text], to appear in Nagoya Math. J.; https://doi.org/10.1017/nmj.2018.36 ], we proved that the value of [Formula: see text] of the generalized hypergeometric function is a [Formula: see text]-linear combination of log of algebraic numbers if rational numbers [Formula: see text] satisfy a certain condition. In this paper, we present a method to obtain an explicit description of it.

Hecke operators on certain subspaces of integral weight modular forms

Recent works of F. G. Garvan ([Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank, Int. J. Number Theory6(12) (2010) 281–309; MR2646759 (2011j:05032)]) and Y. Yang ([Congruences of the partition function, Int. Math. Res. Not.2011(14) (2011) 3261–3288; MR2817679 (2012e:11177)] and [Modular forms for half-integral weights on SL 2(ℤ), to appear in Nagoya Math. J.]) concern a certain family of half-integral weight Hecke-invariant subspaces which arise as multiples of fixed odd powers of the Dedekind eta-function multiplied by SL 2(ℤ)-forms of fixed weight. In this paper, we study the image of Hecke operators on subspaces which arise as multiples of fixed even powers of eta multiplied by SL 2(ℤ)-forms of fixed weight.

Uniqueness for Meromorphic Functions Concerning Differential Polynomials

Using the notion of weighted sharing of values which was introduced by I. Lahiri [Nagoya Math. J. 161: 193–206, 2001], we deal with the uniqueness problem of meromorphic functions concerning differential polynomials and obtain some theorems which not only improve a recent result of W. L. Xiong, W. C. Lin and S. Mori [Sci. Math. Jpn. 62: 305–315, 2005], but also improve and supplement the result of W. C. Lin and H. X. Yi [Complex Var. Theory Appl. 49: 793–806, 2004].

Errata for “L2-boundedness of the Cauchy transform on smooth non-Lipschitz curves”, Nagoya Math J. 130 (1993), 123–147

Note that an extra condition of symmetry of f, namely, the boundedness of f(x) = f(− x) is added to the hypothesis of the Lemma. Lemma 2.1 was used in three places in the paper to prove that the functions F, G*, and f given in pages 140, 141, and 144, respectively, belong to BMO. It can be checked by a standard argument that these functions satisfy the symmetry condition.

Addendum to “On the Bergman kernel of hyperconvex domains”, Nagoya Math. J. 129 (1993), 43–52

0. In [0-1] it was proved that for any bounded hyperconvex domain D in C2 the Bergman kernel function K(z, w) of D satisfiesIn case n ═ 1, this is due to a behavior of sublevel sets of the Green function. The general case then follows by the extendability of L2 holomorphic functions.

Correction to “Isospectral surfaces of small genus,” (Nagoya Math. J. Vol. 107 (1987), 13-24)

It was brought to our attention by Zoran Luicic and Milica Stojanovic, via Peter Gilkey, that some of the diagrams in our paper are not correct.The particular problems are the gluing diagrams for the pair of isospectral surfaces of genus 4, which occur on page 20. It is easy to check that the gluing diagrams given there give rise to a surface of the wrong genus. The problem arose because of carelessness in some of the identifications of some of the edges of the fundamental domain.

Correction to “Gromov’s convergence theorem and its application” (Nagoya Math. J. Vol. 100 (1985), 11–48)

In the proof of Lemma 12.2, the following inequality (p. 43, line 11) is incorrect.This should be corrected as follows.