Günter Meinardus (1926–2007)

Meinardus’ theorem on weighted partitions: extensions and a probabilistic proof

The number cn of weighted partitions of an integer n, with parameters (weights) bk, k ≥ 1, is given by the generating function relationship ∑∞ n=0 cnz n = ∏∞ k=1(1− zk)−bk . Meinardus(1954) established his famous asymptotic formula for cn, as n → ∞, under three conditions on power and Dirichlet generating functions for the sequence bk. We give a probabilistic proof of Meinardus’ theorem with we...

On Meinardus' examples for the conjugate gradient method

The Conjugate Gradient (CG) method is widely used to solve a positive definite linear system Ax = b of order N . It is well-known that the relative residual of the kth approximate solution by CG (with the initial approximation x0 = 0) is bounded above by 2 h ∆κ + ∆ −k κ i−1 with ∆κ = √ κ + 1 √ κ− 1 , where κ ≡ κ(A) = ‖A‖2‖A−1‖2 is A’s spectral condition number. In 1963, Meinardus (Numer. Math.,...

Developments in the Khintchine-Meinardus Probabilistic Method for Asymptotic Enumeration

A theorem of Meinardus provides asymptotics of the number of weighted partitions under certain assumptions on associated ordinary and Dirichlet generating functions. The ordinary generating functions are closely related to Euler’s generating function ∏∞ k=1 S(z k) for partitions, where S(z) = (1 − z)−1. By applying a method due to Khintchine, we extend Meinardus’ theorem to find the asymptotics...

Convex Polytopes : Examples and Conjectures by Günter M

Data Uncertainty and the Role of Money as an Information Variable for Monetary Policy ∗ , † Günter Coenen

In this study, we perform a quantitative assessment of the role of money as an indicator variable for monetary policy in the euro area. We document the magnitude of revisions to euro area-wide data on output, prices, and money, and find that monetary aggregates have a potentially significant role in providing information about current real output. We then proceed to analyze the information cont...