The purpose of the present manuscript is to survey some of the main ideas that appear in recent research of the author on the topic of applying anabelian geometry to construct a “global multiplicative subspace”— i.e., an analogue of the well-known (local) multiplicative subspace of the Tate module of a degenerating elliptic curve. Such a global multiplicative subspace is necessary to apply the ...

there are several kinds of hyperrings, for example, krasnerhyperrings, multiplicative hyperring, general hyperrings and$h_v$-rings. in a multiplicative hyperring, the multiplication isa hyperoperation, while the addition is a binary operation. in this paper, the notion of derivation on multiplicative hyperrings is introduced and some related properties are investigated. {bf keywords:} multiplic...

Using Serre’s adelic interpretation of the cohomology, we develop “differential and integral calculus” on an algebraic curve X over an algebraically closed constant field k of characteristic zero, define an algebraic analogs of additive and multiplicative multi-valued functions on X, and prove corresponding generalized residue theorem and A. Weil reciprocity law. Using the representation theory...

Granville and Soundararajan have recently suggested that a general study of multiplicative functions could form the basis of analytic number theory without zeros of L-functions; this is the so-called pretentious view of analytic number theory. Here we study multiplicative functions which arise from the arithmetic of number fields. For each finite Galois extension K/Q, we construct a natural cla...

The convergence of the multiplicative multisplitting-type method for solving the linear complementarity problem with an H-matrix is discussed using classical and new results from the theory of splitting. This directly results in a sufficient condition for guaranteeing the convergence of the multiplicative multisplitting method. Moreover, the multiplicative multisplitting method is applied to th...

0.1. The kinds of homotopy theories under consideration in this paper are Waldhausen ∞-categories [2, Df. 2.7]. (We employ the quasicategory model of∞-categories for technical convenience.) These are ∞-categories with a zero object and a distinguished class of morphisms (called cofibrations or ingressive morphisms) that satisfies the following conditions. (0.1.1) Any equivalence is ingressive. ...