We study a variant of multiple zeta values of level 2, which forms a subspace of the space of alternating multiple zeta values. This variant, which is regarded as the ‘shuffle counterpart’ of Hoffman's ‘odd variant’, exhibits nice properties such as duality, shuffle product, parity results, etc., like ordinary multiple zeta values. We also give some conjectures on relations between our values, Hoffman's values, and multiple zeta values.

Let $\{x_n\}$ be a sequence of rational numbers greater than one such that $x_{n+1} \geq x^2_n$ for all sufficiently large $n$ and let $\varepsilon_n \in \{-1,1\}$. Under certain growth conditions on the denominators of $x_{n+1}/x^2_n$ we prove that the irrationality exponent of the number $\sum^{\infty}_{n=1} \varepsilon_n/x_n$ is equal to $\limsup_{n\to\infty}(\log x_{n+1}/\log x_n)$.

We use generalised Zeckendorf representations of natural numbers to investigate mixing properties of symbolic dynamical systems. The systems we consider consist of bi-infinite sequences associated with so-called random substitutions. We focus on random substitutions associated with the Fibonacci, tribonacci and metallic mean numbers and take advantage of their respective numeration schemes.

The Navier-Stokes equations and the elastic wave equations are considered in homogeneous and isotropic spacetimes. The energy estimates of the equations are constructed. The effects of the spatial expansion and contraction are characterized by the dissipative and anti-dissipative properties of the equations. The Cauchy problem for the Navier-Stokes equations is considered under the constant density.

We prove that for all transitive piecewise monotonic maps, the following conditions are equivalent:

(1) All ergodic measures on $[0,1]$ are approximated by periodic ones;

(2) All invariant measures on coding spaces are approximated by periodic ones;

(3) All ergodic measures on coding spaces are approximated by periodic ones.

If we further assume that the map is piecewise increasing and either right or left continuous, then the following condition is also equivalent to (1)–(3).

(4) All invariant measures on $[0,1]$ are approximated by periodic ones.

We also construct an example of a piecewise decreasing and right continuous map which satisfies (1)–(3), but does not satisfy (4).