Let E be an optimal elliptic curve of conductor N , such that the L-function LE(s) of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split. The Gross-Zagier theorem gives a formula that expresses the Birch and Swinnerton-Dyer conjectural order the Shafarevich-Tate group of E over K as a rational number. We extract an integer factor from...

indeed has $D=b¥cdot c^{*}$ for $b^{*}=(0,¥ldots,0,1)$ and $c^{*}$ with entries $¥frac{1}{2}(¥beta_{k}-a_{k})$ . (As Jan Willems once pointed out, all the entries are constants; however, the zeros and ones are stiff structure constants, while only the last row has “soft” parameters, to be encompassed by a rank-one control matrix.) Section 1 presents a canonic decomposition of the state space of...

The solution to a general Sylvester equation AX − XB = GF with a low-rank right-hand side is analyzed quantitatively through the Low-rank Alternating-Directional-Implicit method (LR-ADI) with exact shifts. New bounds and perturbation bounds on X are obtained. A distinguished feature of these bounds is that they reflect the interplay between the eigenvalue decompositions of A and B and the right...

The solution to a general Sylvester equation AX−XB = GF ∗ with a low rank righthand side is analyzed quantitatively through Low-rank Alternating-DirectionalImplicit method (LR-ADI) with exact shifts. New bounds and perturbation bounds on X are obtained. A distinguished feature of these bounds is that they reflect the interplay between the eigenvalue decompositions of A and B and the right-hand ...

Supplementary material A. Symmetric tensors A.1. Background Let Rd1⇥···⇥dm be the set of d1⇥ · · ·⇥dm real m-order tensors. In this paper, we focus on cubical tensors, i.e., d1 = · · · = dm = d. We denote the set of m-order cubical tensors by Rd m . We denote the elements of M 2 Rdm by Mj1,...,jm , where j1, . . . , jm 2 [d]. Let = [ 1, . . . , m] be a permutation of {1, . . . ,m}. Given M 2 Rd...

In this paper the problem of comparing several treatments with a control in a one-way repeated measures design is considered. Multiple testing procedures based on rank transformation data are proposed for determining which treatments are more effective than the control. The results of a Monte Carlo level and power study are presented.