On Witten multiple zeta-functions associated with semisimple Lie algebras IV

In our previous work, we established the theory of multi-variable Witten zeta-functions, which are called the zeta-functions of root systems. We have already considered the cases of types $A_2$, $A_3$, $B_2$, $B_3$ and $C_3$. In this paper, we consider the case of $G_2$-type. We define certain analogues of Bernoulli polynomials of $G_2$-type and study the generating functions of them to determine the coefficients of Witten's volume formulas of $G_2$-type. Next we consider the meromorphic continuation of the zeta-function of $G_2$-type and determine its possible singularities. Finally, by using our previous method, we give explicit functional relations for them which include Witten's volume formulas.Comment: 22 pag

Strong first order S-ROCK methods for stochastic differential equations

Explicit stochastic Runge–Kutta (SRK) methods are constructed for non-commutative Itô and Stratonovich stochastic differential equations. Our aim is to derive explicit SRK schemes of strong order one, which are derivative free and have large stability regions. In the present paper, this will be achieved by embedding Chebyshev methods for ordinary differential equations in SRK methods proposed by Rößler (2010). In order to check their convergence order, stability properties and computational efficiency, some numerical experiments will be performed

Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise

A new class of third order Runge-Kutta methods for stochastic differential equations with additive noise is introduced. In contrast to Platen's method, which to the knowledge of the author has been up to now the only known third order Runge-Kutta scheme for weak approximation, the new class of methods affords less random variable evaluations and is also applicable to SDEs with multidimensional noise. Order conditions up to order three are calculated and coefficients of a four stage third order method are given. This method has deterministic order four and minimized error constants, and needs in addition less function evaluations than the method of Platen. Applied to some examples, the new method is compared numerically with Platen's method and some well known second order methods and yields very promising results.Comment: Two further examples added, small correction

Observation of the Ettingshausen effect in quantum Hall systems

Evidence of the Ettingshausen effect in the breakdown regime of the integer quantum Hall effect has been observed in a GaAs/AlGaAs two-dimensional electron system. Resistance of micro Hall bars attached to both edges of a current channel shows remarkable asymmetric behaviors which indicate an electron temperature difference between the edges. The sign of the difference depends on the direction of the electric current and the polarity of the magnetic field. The results are consistent with the recent theory of Akera.Comment: 4 pages, 6 figures, submitted to Phys. Rev.