and Applied Analysis 3 F2 F t, xn, . . . , x0 W t, x0 − H t, xn, . . . , x0 , for every t ∈ Z, W,H are continuously differentiable in x0 and xn, . . . , x0, respectively. Moreover, there is a bounded set J ⊂ Z such that H t, xn, . . . , x0 ≥ 0; 2.2 F3 There is a constant μ > p such that 0 < μW t, x0 ≤ W ′ 2 t, x0 x0, ∀ t, x0 ∈ Z × R \ {0} ; 2.3 F4 H t, 0, . . . , 0 ≡ 0, and there is a constant ...

The solution of many physical evolution equations can be expressed as an exponential of two or more operators. Approximate solutions can be systematically derived by decomposing the exponential in a product form. For time-reversible equations, such as the Hamilton or the Schrödinger equation, it is immaterial whether the decomposition coefficients are positive or negative. For timeirreversible ...

This paper describes a technique to construct Weyl group multiple Dirichlet series. Such series were first introduced in [BBC06], which also described a heuristic means to define, analytically continue, and prove functional equations for a family of Dirichlet series in several complex variables. Several subsequent papers have dealt with the problem of making the heuristic definitions precise an...

Let G be a group generated by r elements g1, g2, . . . , gr. Among the reduced words in g1, g2, . . . , gr of length n some, say γn, represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of γ2n has a limit, called the cogrowth exponent with respect to generators g1, g2, . . . , gr. We show by analytic methods that the numbers γn vary regularl...