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We know at least two ways to generalize multiple zeta(-star) values (MZ(S)Vs) which are q-analogue and t-interpolation. The q-analogue of MZVs was introduced by Bradley (2005), Zhao(2007), etc. On the other hand, polynomials interpolating MZVs and MZSVs using a parameter t were introduced by Yamamoto (2013). In this talk, we consider such two generalizations at the same time, that is, we compose polynomials interpolating q-MZVs and q-MZSVs using a parameter t which are reduced to q-MZVs as t=0 and t-MZVs as q to 1. Then we introduce algebraic setup and some relations for this new MZVs.

We propose new energy dissipative characteristic numerical methods for the approximation of the diffusive Oldroyd-B equations, that are based either on the finite element or finite difference discretization. We prove energy stability of both schemes and illustrate their behaviour on a series of numerical experiments. Using both the diffusive model and the logarithmic transformation of the elastic stress we are able to obtain methods that converge as mesh parameter is refined.This work has been done in the cooperation with M. Tabata, H. Notsu (Waseda University, Tokyo) and supported by the DFG International Reseach Training Group “Mathematical Fluid Dynamics”.

In this talk we will introduce (classical) multiple zeta values (MZVs) and relations among them. Our motivation to investigate relations among MZVs often comes from the dimension conjecture, which has been established by D. Zagier in 1994. We summarize several known relations by drawing a diagram for their implications. Moreover we give concrete statements of the derivation relation, the quasi-derivation relation, and the cyclic sum formula under the usual algebraic setup.

Historically, the k-fold Euler/Zagier sums has attracted specialists and nonspecialists alike with its lovely evaluations. Much the same can be said for multiple zeta (or zeta star) values (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, by using the method of multiple integral representations of series, we establish some expressions of series involving classical harmonic numbers and multiple zeta values.

Kaneko and Zagier recently defined new multiple zeta values called finite multiple zeta values. After we review the dimension conjecture for finite multiple zeta values, we give some known facts including sum formula (by Saito-W('15)) and Bowman-Bradley type theorem (by Saito-W('16)) for finite multiple zeta values. )

In this talk we discuss on a huge class (and conjecturally all) of relations called Kawashima relation for MZVs and its applications. We overview Kawashima's work first. Then we prove algebraically that the duality formula, the quasi-derivation relation, and the cyclic sum formula are included in (the linear part of) Kawashima relation.

In this talk, we introduce some recent results on Heegner points and its application to arithmetic of elliptic curves.

We consider the existence of spatially inhomogeneous time-periodic orbit in a reaction-diffusion population model with spatial-temporal nonlocal delayed growth rate and Dirichlet boundary condition. When the dispersal kernel is of strong or weak type, the scalar reaction-diffusion equation with distributed delay is converted into a system of two or three reaction-diffusion equations without delay. We prove the existence of periodic orbits for the system which are equivalent to periodic orbits for the original scalar model.